Abstract
Given a Hermitian line bundle \(L\rightarrow M\) over a closed, oriented Riemannian manifold M, we study the asymptotic behavior, as \(\epsilon \rightarrow 0\), of couples \((u_\epsilon ,\nabla _\epsilon )\) critical for the rescalings
of the self-dual Yang–Mills–Higgs energy, where u is a section of L and \(\nabla \) is a Hermitian connection on L with curvature \(F_{\nabla }\). Under the natural assumption \(\limsup _{\epsilon \rightarrow 0}E_\epsilon (u_\epsilon ,\nabla _\epsilon )<\infty \), we show that the energy measures converge subsequentially to (the weight measure \(\mu \) of) a stationary integral \((n-2)\)-varifold. Also, we show that the \((n-2)\)-currents dual to the curvature forms converge subsequentially to \(2\pi \Gamma \), for an integral \((n-2)\)-cycle \(\Gamma \) with \(|\Gamma |\le \mu \). Finally, we provide a variational construction of nontrivial critical points \((u_\epsilon ,\nabla _\epsilon )\) on arbitrary line bundles, satisfying a uniform energy bound. As a byproduct, we obtain a PDE proof, in codimension two, of Almgren’s existence result for (nontrivial) stationary integral \((n-2)\)-varifolds in an arbitrary closed Riemannian manifold.
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1 Introduction
A level set approach for the variational construction of minimal hypersurfaces was born from the work of Modica–Mortola [30], Modica [29], and Sternberg [34]. Starting from a suggestion by De Giorgi [12], they highlighted a deep connection between minimizers \(u_\epsilon :M\rightarrow {\mathbb {R}}\) of the Allen–Cahn functional
and two-sided minimal hypersurfaces in M, showing essentially that the functionals \(F_{\epsilon }\) \(\Gamma \)-converge to (\(\frac{4}{3}\) times) the perimeter functional on Caccioppoli sets. Several years later, Hutchinson and Tonegawa [19] initiated the asymptotic study of critical points \(v_{\epsilon }\) of \(F_{\epsilon }\) with bounded energy, without the energy-minimality assumption. They showed, in particular, that their energy measures concentrate along a stationary, integral \((n-1)\)-varifold, given by the limit of the level sets \(v_{\epsilon }^{-1}(0)\).
These developments, together with the deep regularity work by Tonegawa and Wickramasekera on stable solutions [38], opened the doors to a fruitful min–max approach to the construction of minimal hypersurfaces, providing a PDE alternative to the rather involved discretized min–max procedure implemented by Almgren and Pitts [5, 31] in the setting of geometric measure theory. This promising min–max approach based on the Allen–Cahn functionals was recently developed by Guaraco and Gaspar–Guaraco [14, 16], and has been used successfully to attack some profound questions concerning the structure of min–max minimal hypersurfaces—most notably in Chodosh and Mantoulidis’s work on the multiplicity one conjecture [11].
The initial motivation for this paper is to find, in a similar vein, a natural way to construct minimal varieties of codimension two through PDE methods. Recently, other attempts in this direction have been made by Cheng [10] and the second-named author [33], based on the study of the Ginzburg–Landau functionals
on complex-valued maps \(v:M\rightarrow {\mathbb {C}}\). While the Ginzburg–Landau approach can be employed successfully to produce nontrivial stationary rectifiable \((n-2)\)-varifolds (building on the analysis of [8, 28], and others), and leads to existence results of independent interest for solutions of the Ginzburg–Landau equations, it is not yet known whether the varifolds produced in this way are integral, nor is it known whether the full energies \(F_{\epsilon }(v_{\epsilon })\) of the min–max critical points converge to the mass of the limiting minimal variety in the case \(b_1(M)\ne 0\).
While it is possible that these and other technical difficulties may be overcome with sufficient effort—and establishing integrality in particular remains a fascinating open problem—they point to the deeper fact that the Ginzburg–Landau functionals, though intimately related to the \((n-2)\)-area, do not provide a straightforward regularization of the codimension-two area functional. Indeed, we stress that the Ginzburg–Landau energies should be understood first and foremost as a relaxation of the Dirichlet energy for singular maps to \(S^1\), and while the limiting singularities of critical points may coincide with minimal varieties, the associated variational problems exhibit substantial qualitative differences at both large and small scales.
In the present paper, we consider instead the self-dual Yang–Mills–Higgs energy
and its rescalings (for \(\epsilon \in (0,1]\))
for couples \((u,\nabla )\) consisting of a section u of a given Hermitian line bundle \(L\rightarrow M\), and a metric connection \(\nabla \) on L. Here, the nonlinear potential \(W: L\rightarrow {\mathbb {R}}\) is given by
while \(F_{\nabla }\in \Omega ^2({\text {End}}(L))\) denotes the curvature of \(\nabla \).
For the trivial bundle \(L={\mathbb {C}}\times {\mathbb {R}}^2\) on the plane \(M={\mathbb {R}}^2\), a detailed study of the functional (1.1) and its critical points can be found in the doctoral work of Taubes [35, 36]. In [36], all finite-energy critical points \((u,\nabla )\) of (1.1) in the plane are shown to solve the first order systemFootnote 1
known as the vortex equations—a two-dimensional counterpart of the instanton equations in four-dimensional Yang–Mills theory. In particular, all such solutions \((u,\nabla )\) minimize energy among pairs \((u,\nabla )\) with fixed vortex number
and carry energy exactly \(E(u,\nabla )=2\pi |N|\). In [35], Taubes shows moreover that there exist solutions of (1.4) with any prescribed zero set
which are unique up to gauge equivalence, so that [35, 36] together give a complete classification of finite-energy critical points of (1.1) in the plane.
In [18], Hong, Jost, and Struwe initiate the study of the rescaled functionals (1.2) in the limit \(\epsilon \rightarrow 0\) for line bundles \(L\rightarrow \Sigma \) over a closed Riemann surface \(\Sigma \). The main result of [18] shows that, for solutions \((u_{\epsilon },\nabla _{\epsilon })\) of the rescaled vortex equations (given by replacing \(\frac{1}{2}(1-|u|^2)\) with \(\frac{1}{2\epsilon ^2}(1-|u_\epsilon |^2)\) in (1.4)), the curvature \(*\frac{1}{2\pi }F_{\nabla _{\epsilon }}\) converges as \(\epsilon \rightarrow 0\) to a finite sum of Dirac masses of total mass \(|{\text {deg}}(L)|\), away from which \(\nabla _{\epsilon }\) converges to a flat connection \(\nabla _0\), and \(u_{\epsilon }\) to a unit section \(u_0\) with \(\nabla _0u_0=0\), up to change of gauge. While the authors of [18] focus on the vortex equations over Riemann surfaces, they suggest that the asymptotic analysis of the rescaled functionals \(E_{\epsilon }\) may also yield interesting results in higher dimension, pointing to similarities with the Allen–Cahn functionals for scalar-valued functions.
In the present paper, we develop the asymptotic analysis as \(\epsilon \rightarrow 0\) for critical points of \(E_{\epsilon }\) associated to line bundles \(L\rightarrow M\) over Riemannian manifolds \(M^n\) of arbitrary dimension \(n\ge 2\). The bulk of the paper is devoted to the proof of the following theorem, which describes the limiting behavior as \(\epsilon \rightarrow 0\) of the energy measures
and curvatures \(F_{\nabla _{\epsilon }}\) for critical points \((u_{\epsilon },\nabla _{\epsilon })\) satisfying a uniform energy bound.
Theorem 1.1
Let \(L\rightarrow M\) be a Hermitian line bundle over a closed, oriented Riemannian manifold \(M^n\) of dimension \(n\ge 2\), and let \((u_{\epsilon },\nabla _{\epsilon })\) be a family of critical points for \(E_{\epsilon }\) satisfying a uniform energy bound
Then, as \(\epsilon \rightarrow 0\), the energy measures
converge subsequentially, in duality with \(C^0(M)\), to the weight measure \(\mu \) of a stationary, integral \((n-2)\)-varifold V. Also, for all \(0\le \delta <1\),
in the Hausdorff topology. The \((n-2)\)-currents dual to the curvature forms \(\frac{1}{2\pi }F_{\nabla _{\epsilon }}\) converge subsequentially to an integral \((n-2)\)-cycle \(\Gamma \), with \(|\Gamma |\le \mu \).
As will be clear from the proofs, orientability will be assumed only to show the statement concerning the current \(\Gamma \).
Roughly speaking, Theorem 1.1 says that the energy of the critical points concentrates near the zero sets \(u_{\epsilon }^{-1}(0)\) of \(u_{\epsilon }\) as \(\epsilon \rightarrow 0\), which converge to a (possibly rather singular) minimal submanifold of codimension two. In the case \(\dim (M)=3\), for instance, it follows from the results above and work of Allard and Almgren [3] that energy concentrates along a stationary geodesic network with integer multiplicities. The convergence of the curvature, moreover, to an integral cycle Poincaré dual to \(c_1(L)\), with mass bounded above by \(\lim _{\epsilon \rightarrow 0}E_{\epsilon }(u_{\epsilon },\nabla _{\epsilon })\), provides a higher dimensional analog to the limiting behavior described in two dimensions by Hong–Jost–Struwe [18].
At first glance, the obvious advantages of Theorem 1.1 over analogous results for the complex Ginzburg–Landau equations (cf., e.g., [8, 28, 33]) are the integrality of the limit varifold V, and the concentration of the full energy measure to V, independent of the topology of M. Indeed, Theorem 1.1 and the analysis leading to its proof align much more closely with the work of Hutchinson and Tonegawa [19] on the Allen–Cahn equations than they do with related results (e.g. [8, 28]) for the complex Ginzburg–Landau equations. The parallels between the analysis presented here and that of the Allen–Cahn equations in [19] are in fact quite striking in places—a point to which we will draw the reader’s attention throughout the paper.
Remark 1.2
We warn the reader, however, that while the qualitative analysis of the Allen–Cahn functionals does not depend on the precise choice of the double-well potential W, the analysis of the abelian Yang–Mills–Higgs functionals (1.1)–(1.2) seems to depend quite strongly on the choice \(W(u)=\frac{1}{4}(1-|u|^2)^2\). Indeed, already in two dimensions, replacing W with a potential \(W_{\lambda }(u):=\frac{\lambda }{4}(1-|u|^2)^2\) for some \(\lambda \ne 1\) yields a dramatically different qualitative behavior, breaking the symmetry which leads to the first-order equations (1.4), and introducing interactions between disjoint components of the zero set (see, e.g., [21, Chapters I–III]). This should serve as one indication that the analysis of the abelian Higgs model is somewhat more delicate than that of related semilinear scalar equations, in spite of the strong parallels.
To get some idea of the role played by gauge invariance, note that unit sections of a Hermitian line bundle are indistinguishable up to change of gauge (when no preferred connection has been selected) and, for a given unit section u of L, one can always choose locally a connection with respect to which u appears constant. Thus, while most of the energy of solutions \(v_{\epsilon }\) to the complex Ginzburg–Landau equations falls on annular regions—relatively far from the zero set—where \(v_{\epsilon }\) resembles a harmonic \(S^1\)-valued map, the energy \(e_{\epsilon }(u_{\epsilon },\nabla _{\epsilon })\) of a critical pair \((u_{\epsilon },\nabla _{\epsilon })\) for the abelian Yang–Mills–Higgs energy instead concentrates near the zero set \(u_{\epsilon }^{-1}(0)\), with \(|\nabla _{\epsilon }u_{\epsilon }|\) vanishing rapidly outside this region.
Of course, the results of Theorem 1.1 would be of limited interest if nontrivial critical points \((u_{\epsilon },\nabla _{\epsilon })\) could be found only in a few special settings. After completing the proof of Theorem 1.1, we therefore establish the following general existence result, showing that nontrivial families satisfying the hypotheses of Theorem 1.1 arise naturally on any line bundle (including, importantly, the trivial bundle) over any Riemannian manifold \(M^n\), from variational constructions.
Theorem 1.3
For any Hermitian line bundle \(L\rightarrow M\) over an arbitrary closed base manifold \(M^n\), there exists a family \((u_{\epsilon },\nabla _{\epsilon })\) satisfying the hypotheses of Theorem 1.1, with nonempty zero sets \(u_{\epsilon }^{-1}(0)\ne \varnothing \). In particular, the energy \(\mu _{\epsilon }\) of these families concentrates (subsequentially) on a nontrivial stationary integral \((n-2)\)-varifold V as \(\epsilon \rightarrow 0\).
For nontrivial bundles \(L\rightarrow M\), this follows from a fairly simple argument, showing that the minimizers \((u_{\epsilon },\nabla _{\epsilon })\) of \(E_{\epsilon }\) satisfy uniform energy bounds as \(\epsilon \rightarrow 0\). For these energy-minimizing solutions, we expect moreover that the limiting minimal variety 
, i.e. the weight measure |V| of V, coincides with the weight measure \(|\Gamma |\) of the limiting \((n-2)\)-cycle \(\Gamma =\lim _{\epsilon \rightarrow 0} *\frac{1}{2\pi }F_{\nabla _{\epsilon }}\), and that \(\Gamma \) minimizes \((n-2)\)-area in its homology class. While we do not take up this question here, we believe that it would be interesting to study the convergence of the functionals (1.2) to the \((n-2)\)-area functional in a \(\Gamma \)-convergence framework. Let us mention that an asymptotic study for minimizers of the Ginzburg–Landau functional, on a domain with boundary, was successfully carried out by Lin and Rivière [27], who were able to identify the concentration measure with the weight of an integral current. (See also [1, 22] for related \(\Gamma \)-convergence results in that setting.)
Remark 1.4
We remark that a very special class of minimizers for \(E_{\epsilon }\) are given by solutions \((u_{\epsilon },\nabla _{\epsilon })\) of the first-order vortex equations in Kähler manifolds \((M^{2n},\omega _K)\) of higher dimension; these generalize the system (1.4) from the two-dimensional setting by replacing \(*F_{\nabla }\) in (1.4) by the inner product \(\langle F_{\nabla },\omega _K\rangle \) with the Kähler form \(\omega _K\), and requiring additionally that \(F_{\nabla }^{0,2}=0\). As in the two-dimensional setting, solutions of this first-order system minimize the energy \(E_{\epsilon }\) in appropriate line bundles on Kähler manifolds, and it was shown by BradlowFootnote 2 [9] that the moduli space of solutions corresponds to the space of complex subvarieties in M (of complex codimension one) via the zero locus \((u_{\epsilon },\nabla _{\epsilon })\mapsto u_{\epsilon }^{-1}(0)\).
In particular, the zero loci \(u_{\epsilon }^{-1}(0)\) in this case are already area-minimizing subvarieties, before passing to the limit \(\epsilon \rightarrow 0\). Note that the analysis of the vortex equations plays a key role in the study of Seiberg–Witten invariants of Kähler surfaces [39], and a similar analysis figures crucially into Taubes’s work relating the Seiberg–Witten and Gromov–Witten invariants of symplectic four-manifolds [37]. For a concise introduction to the higher-dimensional vortex equations and connections to Seiberg–Witten theory, we refer the interested reader to the survey [13] by García–Prada.
For the trivial bundle \(L\cong {\mathbb {C}}\times M\), we prove Theorem 1.3 by applying min–max methods to the functionals (1.2), to produce nontrivial families \((u_{\epsilon },\nabla _{\epsilon })\) satisfying a uniform energy bound as \(\epsilon \rightarrow 0\). While we consider only one min–max construction in the present paper, we remark that many more may be carried out in principle, due to the rich topology of the space
where \({\mathcal {G}}:={\text {Maps}}(M,S^1)\) is the gauge group. Indeed, on a closed oriented manifold M, one can show that the homotopy groups \(\pi _i({\mathcal {M}})\) are given by
it may be of interest to note that these are isomorphic to the homotopy groups of the space \({\mathcal {Z}}_{n-2}(M;{\mathbb {Z}})\) of integral \((n-2)\)-cycles in M, as computed by Almgren [4].
As an application of Theorem 1.3, we obtain a new proof of the existence of stationary integral \((n-2)\)-varifolds in an arbitrary Riemannian manifold—a result first proved by Almgren in 1965 [5] using a powerful, but rather involved geometric measure theory framework. As already mentioned, similar constructions for the Allen–Cahn equations have been carried out successfully by Guaraco [16] and Gaspar–Guaraco [14], yielding new proofs of the existence of minimal hypersurfaces of optimal regularity, and leading to other recent breakthroughs in the min–max theory of minimal hypersurfaces (e.g., [11]).
In [11, 16] (building on results of [38]), the stability properties of the min–max critical points for the Allen–Cahn functionals play a central role in controlling the regularity and multiplicity of the limit hypersurface. To obtain an improved understanding of min–max families \((u_{\epsilon },\nabla _{\epsilon })\) and the associated minimal varieties in the abelian Higgs setting, it would likewise be very interesting to refine the conclusions of Theorem 1.1 under the assumption that the families \((u_{\epsilon },\nabla _{\epsilon })\) satisfy a uniform Morse index bound as \(\epsilon \rightarrow 0\). We hope to take up this line of investigation in future work.
1.1 Organization of the paper
In Sect. 2 we fix notation and record some basic properties satisfied by critical pairs \((u,\nabla )\) for the energies \(E_{\epsilon }\).
In Sect. 3, we record some useful Bochner identities for the gauge-invariant quantities \(|u|^2\), \(|F_{\nabla }|^2\), and \(|\nabla u|^2\), and use them to establish an initial rough estimate on \(\xi _\epsilon :=\epsilon |F_\nabla |-\frac{(1-|u|^2)}{2\epsilon }\), whose role should be compared to that of the discrepancy function in the Allen–Cahn setting. Under suitable assumptions on the curvature of M, the fact that \(\xi _\epsilon \le 0\) follows quickly from the aforementioned Bochner identities and the maximum principle. Without the curvature assumptions, some nontrivial additional work is required to obtain the pointwise upper bound \(\xi _\epsilon \le C(M,E_\epsilon (u,\nabla ))\). This estimate is the key ingredient to obtain the sharp \((n-2)\)-monotonicity of the energy, and relies on the specific choice of coupling constants appearing in the self-dual Yang–Mills–Higgs functionals.
In Sect. 4 we derive the stationarity equation for inner variations, from which an obvious \((n-4)\)-monotonicity property of the energy follows rather immediately. Using our rough initial bounds on \(\xi _\epsilon \) from Sect. 3, we deduce an intermediate \((n-3)\)-monotonicity; we use this to reach the pointwise bound \(\xi _\epsilon \le C(M,E_\epsilon (u,\nabla ))\), from which we finally infer the sharp \((n-2)\)-monotonicity.
In Sect. 5 we show that, similar to the Allen–Cahn setting, the energy density \(e_{\epsilon }(u,\nabla )\) decays exponentially away from the set \(u^{-1}(0)\)—more precisely, away from \(\{|u|^2\ge 1-\beta _d\}\) for some \(\beta _d\) independent of \(\epsilon \).
Section 6, which constitutes the main part of the paper, contains an initial description of the limiting varifold, showing that it is stationary, \((n-2)\)-rectifiable, and has a lower density bound on the support. Then we establish its integrality with a blow-up analysis, employing the estimates from the preceding sections to reduce the problem to a statement for entire planar solutions, already contained in the work of Jaffe and Taubes [21]. We then use this analysis to show that the level sets \(u_\epsilon ^{-1}(0)\) converge to the support of V in the Hausdorff topology, and conclude the section with a discussion of the asymptotics for the curvature forms \(\frac{1}{2\pi }F_{\nabla _{\epsilon }}\).
In Sect. 7, we show that \(E_\epsilon \) satisfies a variant of the Palais–Smale property on suitable function spaces, allowing us to produce critical points via classical min–max methods. We provide a variational construction to get nontrivial critical points satisfying the assumptions of our main theorem, with energy bounded from above and below, both for nontrivial and trivial line bundles.
Finally, the “Appendix” addresses the issue of showing regularity of critical points, as obtained from Sect. 7, when they are read in a local or global Coulomb gauge.
2 The Yang–Mills–Higgs equations on U(1) bundles
Let M be a closed, oriented Riemannian manifold, and let \(L\rightarrow M^n\) be a complex line bundle over M, endowed with a Hermitian structure \(\langle \cdot ,\cdot \rangle \). Denote by \(W: L\rightarrow {\mathbb {R}}\) the nonlinear potential
For a Hermitian connection \(\nabla \) on L, a section \(u\in \Gamma (L)\) and a parameter \(\epsilon >0\), denote by \(E_{\epsilon }(u,\nabla )\) the scaled Yang–Mills–Higgs energy
where \(F_{\nabla }\) is the curvature of \(\nabla \). Throughout, we will identify the curvature \(F_{\nabla }\) with a closed real two-form \(\omega \) via
In computing inner products for two-forms, we follow the convention
with respect to a local orthonormal basis \(\{e_j\}_{j=1}^n\) for TM.
Note that \(E_\epsilon \) enjoys the U(1) gauge invariance
for any (smooth) \(\theta :M\rightarrow {\mathbb {R}}\). More generally, we have
for any \(\varphi :M\rightarrow S^1\), identifying \(S^1\) with the unit circle in \({\mathbb {C}}\).
It is easy to check that the smooth pair \((u,\nabla )\) gives a critical point for the energy \(E_{\epsilon }\), with respect to smooth variations, if and only if it satisfies the system
We denote \(\Delta _H=dd^*+d^*d\) the usual positive definite Hodge Laplacian on differential forms and note that, in our convention, the adjoint to \(d:\Omega ^1(M)\rightarrow \Omega ^2(M)\) is
Since the curvature form \(\omega \) is closed, taking the exterior derivative of (2.5) gives
i.e.,
where
For future reference, we record the simple bound
To confirm (2.7), fix \(x\in M\) and note that the linear map \(\nabla u(x):T_xM\rightarrow L_x\) has a kernel of dimension at least \(n-2\). Take an orthonormal basis \(\{e_j\}\) of \(T_xM\) such that \(e_j\in {\text {ker}}\nabla u(x)\) for \(j>2\). We compute at x that
which gives (2.7).
3 Bochner identities and preliminary estimates
From the Eqs. (2.6) and (2.4), we apply the standard Bochner–Weitzenböck formulas to obtain some identities which will play a central role in our analysis. For the curvature two-form \(\omega \), it will be useful to record the Bochner identity
where D is the Levi–Civita connection and \({\mathcal {R}}_2\) denotes the Weitzenböck curvature operator for two-forms on the base Riemannian manifold M. For any \(\delta >0\) we have
Since \(|D|\omega ||^2\le |D\omega |^2\), (3.1) implies
Dividing by \((|\omega |^2+\delta ^2)^{1/2}\) and letting \(\delta \rightarrow 0\), we obtain
in the distributional sense (and classically on \(\{|\omega |>0\}\)). Note that, by (2.7), the relation (3.2) also gives us the cruder subequation
For the modulus \(|u|^2\) of the Higgs field u, we record
and observe that a simple application of the maximum principle yields the pointwise bound
For the energy density \(|\nabla u|^2\) of the Higgs field u, we see that
where at \(p\in M\) we let \({\mathcal {R}}_1(\nabla u,\nabla u)={\text {Ric}}(e_i,e_j)\langle \nabla _{e_i}u,\nabla _{e_j}u\rangle \) and \(\nabla ^2_{e_i,e_j}u=\nabla _{e_i}(\nabla _{e_j}u)\), for any local orthonormal frame \(\{e_i\}_{i=1}^n\) with \(De_i(p)=0\).
Next, we introduce the function
and combine (3.3) with (3.4) to see that
If \({\mathcal {R}}_2>0\), we can actually replace the term \(-\epsilon \Vert {\mathcal {R}}_2^-\Vert _{L^\infty }|\omega |\) with \(c\epsilon |\omega |\), for some positive constant \(c=c(M)\); from a simple application of the maximum principle, in this case we get \(\xi _{\epsilon }\le 0\) everywhere on M, and consequently (cf. [21, Theorem III.8.1])
This balancing of the Yang–Mills and potential terms, which should be compared with Modica’s gradient estimate in the asymptotic analysis of the Allen–Cahn equations (cf. [19, Proposition 3.3]), will play a key role in our analysis, allowing us to upgrade the obvious \({(n-4)}\)-monotonicity typical of Yang–Mills–Higgs problems to the much stronger \({(n-2)}\)-monotonicity \(\frac{d}{dr}(r^{2-n}\int _{B_r}e_{\epsilon }(u,\nabla ))\ge 0\).
Remark 3.1
We remark that the analog of the identity \(\Delta \xi _{\epsilon }\ge \epsilon ^{-2}|u|^2\xi _{\epsilon }-\epsilon \Vert {\mathcal {R}}_2^-\Vert _{L^{\infty }}|\omega |\)—and, consequently, the sharp \((n-2)\)-monotonicity result—fails for choices of coupling constants other than those corresponding to the self-dual Yang–Mills–Higgs functionals considered here.
Without the positive curvature assumption, we may still employ the subequation
to obtain strong estimates for the positive part \(\xi _{\epsilon }^+\) of \(\xi _{\epsilon }\). To begin, denote by G(x, y) the nonnegative Green’s function for the Laplacian on M, unique up to additive constant, so that \(\Delta _xG(x,y)=\frac{1}{{\text {vol}}(M)}-\delta _y\), and set
so that
Taking \(C'\) to be the constant appearing in (3.7), for the difference \(\xi _{\epsilon }-C'h_{\epsilon }\) we then have
Observe that the \(L^1\) norm of \(\xi _\epsilon -C'h_\epsilon \) is bounded by the energy:
(Where the constant C(M) may of course change from line to line.)
Integrating (3.10) against the positive part \(\zeta :=(\xi _{\epsilon }-C'h_{\epsilon })^+\) and bounding \(\Vert \epsilon F_{\nabla }\Vert _{L^1}\le C(M)E_{\epsilon }(u,\nabla )^{1/2}\), we get
Applying (3.11), this gives \(\Vert d\zeta \Vert _{L^2}\le C(M)E_{\epsilon }(u,\nabla )\).
Thus, applying Moser iteration, namely integrating (3.10) against powers \(\zeta ^\gamma \) with increasing exponents \(\gamma >1\), we deduce that
As a simple application of (3.12), we note that by definition (3.8) of \(h_{\epsilon }\) and the standard estimate (see, e.g., [7, Section 4.2])
if \(n\ge 3\) (or \(G(x,y)\le -C(M)\log (d(x,y))+C(M)\) if \(n=2\)), we have the \(L^{\infty }\) estimate
(with 2 replacing \(n-1\) when \(n=2\)). If \(n=2\), this inequality and (3.12) give a pointwise bound
In the sequel, we assume \(n\ge 3\) and aim for a similar pointwise bound. We have
Using this in (3.12), we compute at a maximum point for \(|F_{\nabla }|\) to see that
and, by an application of Young’s inequality, it follows that
for any \(\delta \in (0,1)\). Taking \(\delta =\epsilon ^{2/n}\), we arrive at the crude preliminary estimate
where \(\alpha (\epsilon )\rightarrow 0\) as \(\epsilon \rightarrow 0\).
Now, consider the function
By virtue of the preceding estimate for \(\Vert F_{\nabla }\Vert _{L^{\infty }}\), we then see that
pointwise. Appealing once again to (3.4) and (3.3), we see that
so at a point where f achieves its maximum we have
On the other hand, we know that \(|u|^2\ge \frac{\epsilon }{C(1+E_{\epsilon }(u,\nabla )^{1/2})}f\) everywhere, so the preceding computations yield an estimate of the form
provided \(\max f\ge 0\), and we deduce that \(f\le C(M,E_{\epsilon }(u,\nabla ))\) everywhere. Putting all this together, we arrive at the following lemma.
Lemma 3.2
Let \((u,\nabla )\) solve (2.4) and (2.5) on a line bundle \(L\rightarrow M\), and suppose \(E_{\epsilon }(u,\nabla )\le \Lambda \). Then there exist a constant \(C(M,\Lambda )\) and a function \(\alpha (M,\Lambda ,\epsilon )\), with \(\alpha (\epsilon )\rightarrow 0\) as \(\epsilon \rightarrow 0\), such that
In the next section, we will improve the rough preliminary estimate of Lemma 3.2 to a uniform pointwise bound of the form \(\xi _{\epsilon }\le C(M,\Lambda )\), but this will require some additional effort.
4 Inner variations and improved monotonicity
In this section, we derive the inner variation equation for solutions of (2.4)–(2.5), and explore the scaling properties of the energy \(E_{\epsilon }(u_{\epsilon },\nabla _{\epsilon })\) over balls of small radius. Under the assumption that the curvature operator \({\mathcal {R}}_2\) appearing in (3.3) is positive-definite (so that (3.6) holds), the analysis simplifies considerably, leading with little effort to the desired monotonicity of the \((n-2)\)-energy density. Without this curvature assumption, more work is required, first building on the cruder estimates of the preceding section to obtain a uniform pointwise bound for \(\xi _{\epsilon }\).
Fixing notation, with respect to a local orthonormal basis \(\{e_i\}\) for TM, define the (0, 2)-tensors \(\nabla u^*\nabla u\) and \(\omega ^*\omega \) by
Note that \({\text {tr}}(\nabla u^*\nabla u)=|\nabla u|^2\) and \({\text {tr}}(\omega ^*\omega )=2|\omega |^2\). Denote by \(e_{\epsilon }(u,\nabla )\) the energy integrand
The fact that \(d\omega =0\) reads
where D is the Levi–Civita connection of M. Using this identity, it is straightforward to check that
In particular, defining the stress-energy tensor \(T_{\epsilon }(u,\nabla )\) by
for \((u,\nabla )\) solving (2.4) and (2.5) it follows that
meaning that \(\sum _i (D_{e_i}T_\epsilon )(e_i,\cdot )=0\). Integrating (4.4) against a vector field X on some domain \(\Omega \subseteq M\), we arrive at the usual inner-variation equation
where we identify \(T_\epsilon (u,\nabla )\) with a (1, 1)-tensor and denote by \(\nu \) the outer unit normal to \(\Omega \). Taking \(\Omega =B_r(p)\) to be a small geodesic ball of radius r about a point \(p\in M\), and taking \(X={\text {grad}}(\frac{1}{2}d_p^2)\), where \(d_p\) is the distance function to p, (4.5) gives
Now, by the Hessian comparison theorem, we know that
applying this in the relations above, we see that
Setting
it follows from the computations above (temporarily throwing out the additional nonnegative boundary terms) that
At this point, one easily observes that the right-hand side of (4.7) is bounded below by \(\frac{n-4}{r}f(p,r)\), to obtain the monotonicity of the \((n-4)\)-energy density
For general Yang–Mills and Yang–Mills–Higgs problems, this codimension-four energy growth is well known to be sharp (cf., e.g., [32, 40]). For solutions of (2.4) and (2.5) on Hermitian line bundles, however, we show now that this can be improved to (near-) monotonicity of the \((n-2)\)-density \(r^{2-n}f(p,r)\) on small balls, which constitutes a key technical ingredient in the proof of Theorem 1.1.
To begin, we rearrange (4.7), to see that
recalling the notation \(\xi _{\epsilon }:=\epsilon |F_{\nabla }|-\frac{1}{2\epsilon }(1-|u|^2)\). Now, by Lemma 3.2, assuming \(E_{\epsilon }(u,\nabla )\le \Lambda \), we have the pointwise bound
Applying this in our preceding computation for \(\frac{\partial f}{\partial r}\), we deduce that
for some constant \(C''(M,\Lambda )\) and \(0<r<c(M)\). Taking \(\epsilon \) sufficiently small, we arrive next at the following coarse estimate for the \((n-3)\)-energy density, which we will then use to establish an improved bound for \(\xi _{\epsilon }\).
Lemma 4.1
For \(\epsilon \le \epsilon _m(M,\Lambda )\) sufficiently small, we have a uniform bound
Proof
The statement is trivial if \(n=2,3\), so assume \(n\ge 4\). In the preceding computation, take \(\epsilon \le \epsilon _m(M,\Lambda )\) sufficiently small that \(C''\alpha (\epsilon )<\frac{1}{2}\). Then the estimate gives
from which it follows that, for \(0<r<c(M)\),
If \(r^{3-n}f(r)\) has a maximum in (0, c(M)), it follows that \(f(r)\le C r^{n/2}f(r)^{1/2}\) there, and therefore \(r^{3-n}f(r)\le C r^3\le C\). Obviously the desired estimate holds at \(r=0\) and \(r=c(M)\), so (4.8) follows. \(\square \)
With Lemma 4.1 in hand, we can now improve the bounds of Lemma 3.2 to a uniform pointwise estimate, as follows.
Proposition 4.2
Let \((u,\nabla )\) solve (2.4)–(2.5) on a line bundle \(L\rightarrow M\), with the energy bound \(E_{\epsilon }(u,\nabla )\le \Lambda \) and \(\epsilon \le \epsilon _m\). Then there is a constant \(C(M,\Lambda )\) such that
Proof
We can assume \(n\ge 3\), as we already obtained the claim for \(n=2\) in Sect. 3. Recall from that section the function
where G is the nonnegative Green’s function on M. As discussed in Sect. 3, we can deduce from (3.7) a pointwise estimate of the form
Thus, to arrive at the desired bound (4.9), it will suffice to establish a pointwise bound of the same form for \(h_{\epsilon }\).
To this end, recall again that \(G(x,y)\le C(M)d(x,y)^{2-n}\), so that by definition we have
where the last line is a simple application of Young’s inequality. Since the integral \(\int _M d(x,y)^{-n+1/2}\,dy\) is finite, it follows that
On the other hand, by Lemma 4.1, we know that \(r^{3-n}\int _{B_r(x)}e_{\epsilon }(u,\nabla )\le C(M,\Lambda )\) for every r, so we see finally that
as desired. \(\square \)
Applying (4.9) in our original computation for \(f'(r)\), we see now that
In fact, bringing in the extra boundary terms that we have been neglecting, and applying Young’s inequality to the term \(r^{\frac{n-2}{2}}f(r)^{1/2}\), we see that
With this differential inequality in place, a straightforward computation leads us finally to one of our key technical theorems, the monotonicity formula for the \((n-2)\)-density.
Theorem 4.3
Let \((u,\nabla )\) solve (2.4)–(2.5) on a Hermitian line bundle \(L\rightarrow M\), with an energy bound \(E_{\epsilon }(u,\nabla )\le \Lambda \). Then there exist positive constants \(\epsilon _m(M,\Lambda )\) and \(C_m(M,\Lambda )\) such that the normalized energy density
satisfies
for \(0<r<{\text {inj}}(M)\) and \(\epsilon \le \epsilon _m\).
As a simple corollary of the monotonicity result (together with a pointwise bound for \(|\nabla u|\) derived in the following section), we deduce that \((u,\nabla )\) must have positive \((n-2)\)-energy density wherever |u| is bounded away from 1.
Corollary 4.4
(clearing-out) Let \((u,\nabla )\) solve (2.4)–(2.5) on a line bundle \(L\rightarrow M\), with \(E_{\epsilon }(u,\nabla )\le \Lambda \) and \(\epsilon \le \epsilon _m\). Given \(0<\delta <1\), if
with \(x\in M\) and \(\epsilon<r<{\text {inj}}(M)\), then we must have \(|u(x)|>1-\delta \).
Proof
For \(\epsilon \le \epsilon _m\), Theorem 4.3 gives
The gradient bound (5.3) in Proposition 5.1 of the following section gives \(|d|u||\le C\epsilon ^{-1}\). Hence, if \(|u(x)|\le 1-\delta \) then \(|u(y)|<1-\frac{\delta }{2}\) on \(B_{\epsilon \delta /(2C)}(x)\), so that \(1-|u(y)|^2\ge 1-|u(y)|>\frac{\delta }{2}\). We deduce that
Since \({\text {vol}}(B_{\epsilon \delta /(2C)}(x))\) is bounded below by \(c(M,\Lambda ,\delta )\epsilon ^n\), we can choose \({{\widetilde{\eta }}}(M,\Lambda ,\delta )\le {\text {inj}}(M)\) so small that we get a contradiction if \(r,\eta \le {{\widetilde{\eta }}}\). On the other hand, if \(r>{{\widetilde{\eta }}}\) then
Hence, setting \(\eta :=\Big (\frac{{{\widetilde{\eta }}}}{{\text {inj}}(M)}\Big )^{n-2}{{\widetilde{\eta }}}\le {{\widetilde{\eta }}}\), we can reduce to the previous case (replacing r with \({{\widetilde{\eta }}}\)), reaching again a contradiction. \(\square \)
5 Decay away from the zero set
Again, let \((u,\nabla )\) solve (2.4)–(2.5) on a line bundle \(L\rightarrow M\), with the energy bound \(E_{\epsilon }(u,\nabla )\le \Lambda \). In the preceding section, we obtained the pointwise estimate
when \(\epsilon \le \epsilon _m\). As a first step toward establishing strong decay of the energy away from the zero set of u, we show in the following proposition that the full energy density \(e_{\epsilon }(u,\nabla )\) is controlled by the potential \(\frac{W(u)}{\epsilon ^2}\).
Proposition 5.1
For \((u,\nabla )\) as above, we have the pointwise estimates
and
provided \(\epsilon \le \epsilon _d\), for some \(\epsilon _d=\epsilon _d(M,\Lambda )\).
Proof
To begin, let \(C_1=C_1(M,\Lambda )\) be the constant from (5.1), and consider the function
Similar to the proof of Lemma 3.2, observe that \(C_1|u|^2\ge f\) pointwise, by (5.1), while the computations from Sect. 3 give
By (5.1) we have \(|F_\nabla |\le \frac{1}{2\epsilon ^2}+\frac{C_1}{\epsilon }\), so at a positive maximum for f it follows that
so that
(provided \(\max f\ge 0\)), and consequently \(f\le C\epsilon ^{1/2}\) everywhere. As a consequence, at any point, we have either \(f<0\), in which case
or \(f\ge 0\), in which case
In either scenario, we obtain a bound of the desired form (5.2).
To bound \(|\nabla u|^2\), recall from Sect. 3 the identity
In view of the estimate (5.1) for \(|F_{\nabla }|=|\omega |\) and (2.7), we can estimate the term \(2\langle \omega ,\psi (u)\rangle \) from above by
to obtain the existence of \(C_2(M,\Lambda )\) such that
For \(\Delta |\nabla u|\), this then gives
Recalling once again the Eq. (3.4) for \(\Delta \frac{1}{2}|u|^2\), we define
and observe that
We then have
If w has a positive maximum, it follows that
at this maximum point; in particular, we deduce then that
at this point, and see from (5.6) that here
If \(\epsilon \le \epsilon _d(M,\Lambda )\) is small enough, it follows that \(\max w\le C\epsilon \); as a consequence, we check that
completing the proof of (5.3). \(\square \)
As a simple consequence of the estimates in Proposition 5.1, we obtain the following corollary.
Corollary 5.2
There exist constants \(0<\beta _d(M,\Lambda )<1\) and \(C(M,\Lambda )\) such that, for \((u,\nabla )\) as above, we have
on the set \(Z_{\beta _d}(u):=\{|u|^2\ge 1-\beta _d\}\).
Proof
By the formula (3.4) for \(\Delta \frac{1}{2}|u|^2\), we know that
Combining this with the estimate (5.3) for \(|\nabla u|^2\), we then deduce the existence of a constant \({{\widehat{C}}}={{\widehat{C}}}(M,\Lambda )\) such that
By taking \(\beta _d=\beta _d(M,\Lambda )>0\) sufficiently small, we can arrange that
on \(\{|u|^2\ge 1-\beta _d\}\), from which the claimed estimate follows. \(\square \)
Next, we employ the result of Corollary 5.2 to show that the quantity \((1-|u|^2)\) vanishes rapidly away from \(Z_{\beta _d}(u)\) (compare [21, Sections III.7–III.8]).
Proposition 5.3
Let \((u,\nabla )\) be as before, with \(\epsilon \le \epsilon _d\), and define the set
where \(\beta _d(M,\Lambda )\) is the constant provided by Corollary 5.2. Defining \(r: M\rightarrow [0,\infty )\) by
we have an estimate of the form
for some \(C=C(M,\Lambda )\) and \(a_d=a_d(M)>0\).
Proof
Fix a point \(p\in M\), and let \(r=r(p)={\text {dist}}(p,Z_{\beta })\) as above. We can clearly assume \(r(p)<\frac{1}{2}{\text {inj}}(M)\). On the ball \(B_r(p)\), for some constant \(a=a_d>0\) to be chosen later, consider the function
where \(d_p(x):={\text {dist}}(p,x)\). A straightforward computation then gives
for some \(C_1=C_1(M)\). Now, fix some constant \(c_2>0\) to be chosen later, and let
Combining the preceding computation with (5.7), we see that, on \(B_r(p)\),
Choosing \(a=a_d(M)>0\) sufficiently small, we can arrange that \(2a^2+2C_1a\le 1\), so that the above computation gives
On the boundary of the ball \(\partial B_r(p)\), it follows from definition of \(r=r(p)\) that \(|u|^2\ge 1-\beta _d\), and therefore
Taking \(c_2:=\beta _d e^{-ar/\epsilon }\), it then follows that \(f<0\) on \(\partial B_r(p)\), so we can apply the maximum principle with (5.9) to deduce that
Evaluating at p, this gives
so that
as desired. \(\square \)
Combining these estimates with those of Proposition 5.1, we arrive immediately at the following decay estimate for the energy integrand \(e_{\epsilon }(u,\nabla )\).
Corollary 5.4
Defining \(Z_{\beta _d}\) and \(r(p)={\text {dist}}(p,Z_{\beta _d})\) as in Proposition 5.3, there exist \(a_d(M)>0\) and \(C_d(M,\Lambda )\) such that
6 The energy-concentration varifold
This section is devoted to the proof of the main result of the paper, which we recall now.
Theorem 6.1
Let \((u_{\epsilon },\nabla _{\epsilon })\) be a family of solutions to (2.4)–(2.5) satisfying a uniform energy bound \(E_{\epsilon }(u_{\epsilon },\nabla _{\epsilon })\le \Lambda \) as \(\epsilon \rightarrow 0\). Then, as \(\epsilon \rightarrow 0\), the energy measures
converge subsequentially, in duality with \(C^0(M)\), to the weight measure of a stationary, integral \((n-2)\)-varifold V. Also, for all \(0\le \delta <1\),
in the Hausdorff topology. The \((n-2)\)-currents dual to the curvature forms \(\frac{1}{2\pi }\omega _{\epsilon }\) converge subsequentially to an integral \((n-2)\)-cycle \(\Gamma \), with \(|\Gamma |\le \mu \).
6.1 Convergence to a stationary rectifiable varifold
Let \((u_{\epsilon },\nabla _{\epsilon })\) be as in Theorem 6.1, and pass to a subsequence \(\epsilon _j\rightarrow 0\) such that the energy measures \(\mu _{\epsilon _j}\) converge weakly-* to a limiting measure \(\mu \), in duality with \(C^0(M)\).
Note that, for \(0<r<R<{\text {inj}}(M)\), Theorem 4.3 yields
with \(C=C_m\), so approximating R with smaller radii we deduce
and in particular the \((n-2)\)-density
is defined. As a first step toward the proof of Theorem 6.1, we show that this density is bounded from above and below on the support \({\text {spt}}(\mu )\).
Proposition 6.2
There exists a constant \(0<C=C(M,\Lambda )<\infty \) such that
and thus \(C^{-1}\le \Theta _{n-2}(\mu ,x)\le C\) for all \(x\in {\text {spt}}(\mu )\).
Proof
The upper bound follows from (6.1), which gives (when \(R={\text {inj}}(M)\))
To see the lower bound, let \(\beta _d=\beta _d(M,\Lambda )\in (0,1)\) be the constant given by Corollary 5.4, and again set
Let \(\Sigma \) be the set of all limits \(x=\lim _{\epsilon }x_\epsilon \), with \(x_\epsilon \in Z_{\beta _d}(u_\epsilon )\); that is, take
We then claim that
and
Once both (6.3) and (6.4) are established, the lower bound in (6.2) follows immediately.
To establish (6.3), fix some \(p\in M{\setminus } \Sigma \); by definition of \(\Sigma \), there must exist \(\delta =\delta (p)>0\) such that
for all \(\epsilon \) sufficiently small. Applying Corollary 5.4 for all \(x\in B_{\delta }(p)\), we deduce that
In particular, \(p\in M{\setminus } {\text {spt}}(\mu )\), confirming (6.3).
To see (6.4), let \(x\in \Sigma \). Note that, by definition of \(\Sigma \), there exist points \(x_{\epsilon }\in Z_{\beta _d}(u_{\epsilon })\) with \(x_{\epsilon }\rightarrow x\) as \(\epsilon \rightarrow 0\) (along a subsequence). We then see that
and Corollary 4.4 gives \(c(M,\Lambda )\) such that
for \(\epsilon<r<{\text {inj}}(M)\). Since for any \(\delta >0\) we have \(B_r(x_\epsilon )\subseteq {{\overline{B}}}_{r+\delta }(x)\) eventually, it follows that \(\mu ({{\overline{B}}}_{r+\delta }(x))\ge cr^{n-2}\), hence
for \(0<r<{\text {inj}}(M)\), which is (6.4). \(\square \)
With Proposition 6.2 in place, we will invoke a result by Ambrosio and Soner [6] to conclude that the limiting measure \(\mu =\lim _{\epsilon \rightarrow 0}\mu _{\epsilon }\) coincides with the weight measure of a stationary, rectifiable \((n-2)\)-varifold. Recall from Sect. 4 the stress-energy tensors
We record first the following lemma; in its statement, we canonically identify (and pair with each other) tensors of rank (2, 0), (1, 1), and (0, 2), using the underlying metric g.
Lemma 6.3
As \(\epsilon \rightarrow 0\), the tensors \(T_{\epsilon }\) converge (subsequentially) as \({\text {Sym}}(TM)\)-valued measures, in duality with \(C^0(M,{\text {Sym}}(TM))\), to a limit T satisfying
and
Proof
For each \(\epsilon >0\), note that, by definition of \(T_{\epsilon }\), for every continuous vector field \(X\in C^0(M,TM)\) we have
Evaluating (2.3) in an orthonormal basis such that X is a multiple of \(e_1\), we see that \(|\iota _X F_{\nabla _{\epsilon }}|^2\le |F_{\nabla _\epsilon }|^2|X|^2\), while \(|(\nabla _{\epsilon })_X u_{\epsilon }|^2\le |\nabla _{\epsilon } u_{\epsilon }|^2|X|^2\). We deduce that
As an immediate consequence, we see that the uniform energy bound \(E_{\epsilon }(u_{\epsilon },\nabla _{\epsilon })\le \Lambda \) gives a uniform bound on \(\Vert T_{\epsilon }\Vert _{(C^0)^*}\) as \(\epsilon \rightarrow 0\), so we can indeed extract a weak-* subsequential limit \(T\in C^0(M,{\text {Sym}}(TM))^*\), for which (6.7) follows from (6.8).
The stationarity condition (6.5) for the limit T follows from (4.5). It remains to establish the trace inequality (6.6). For this, we simply compute, for nonnegative \(\varphi \in C^0(M)\),
Recalling from Proposition 4.2 that
we then see that
In particular, (6.6) will follow once we show that \(\lim _{\epsilon \rightarrow 0}\int _M e_{\epsilon }(u_{\epsilon },\nabla _{\epsilon })^{1/2}=0\).
But this is straightforward: from Proposition 6.2 we know that for \(0<\delta <{\text {inj}}(M)\) we have
Since \({\text {vol}}(B_{5\delta }(x))\le C(M)\delta ^n\), a simple Vitali covering argument then implies that the \(\delta \)-neighborhood \(B_{\delta }(\Sigma )\) of \(\Sigma \) satisfies a volume bound
With this estimate in hand, we then see that
Fixing \(\delta \) and taking the limit as \(\epsilon \rightarrow 0\), we have \(\mu _{\epsilon }(M{\setminus } B_{\delta }(\Sigma ))\rightarrow 0\). Since \({\text {vol}}(B_{\delta }(\Sigma ))\le C \delta ^2\), we find that
Finally, taking \(\delta \rightarrow 0\), we conclude that \(\int _Me_{\epsilon }(u_{\epsilon },\nabla _{\epsilon })^{1/2}\rightarrow 0\) as \(\epsilon \rightarrow 0\), completing the proof. \(\square \)
Estimate (6.7) says that |T| is absolutely continuous with respect to \(\mu \), so by the Radon–Nikodym theorem we can write the limiting \({\text {Sym}}(TM)\)-valued measure T from Lemma 6.3 as
for some \(L^{\infty }\) (with respect to \(\mu \)) section \(P: M\rightarrow {\text {Sym}}(TM)\). Moreover, it follows from (6.6) and (6.7) that \(-g \le P(x)\le g\) and \({\text {tr}}(P(x))\ge n-2\) at \(\mu \)-a.e. \(x\in M\), so that \(\frac{1}{2\pi }T\) defines in a natural way a generalized \((n-2)\)-varifold in the sense of Ambrosio and Soner, namely a Radon measure on the bundle
We refer the reader to [6, Section 3]. Note that in [6] the authors work in the Euclidean space and require the trace to be equal to \(n-2\) in (6.10); however, the main result on generalized varifolds, namely [6, Theorem 3.8], still holds in our setting. Indeed, in the proof of part (a) of that theorem, the condition \(\sum _{i=1}^{m+1}\lambda _i=m\) that the authors obtain becomes \(\sum _{i=1}^{m+1}\lambda _i\ge m\) in our setting (with \(m=n-2\)), and the constraint \(\lambda _i\le 1\) still ensures the conclusion \(\lambda _i\ge 0\) for all i. Similarly, for part (b), the condition \(\sum _{i=1}^m\lambda _i=m\) has to be replaced by \(\sum _{i=1}^m\lambda _i=m\), and this still implies \(\lambda _i=1\) for all \(i=1,\dots ,m\).
Hence, in view of the stationarity condition (6.5) and the density bounds of Proposition 6.2, we can apply [6, Theorem 3.8(c)] to conclude that \(\frac{1}{2\pi }T\) can be identified with a stationary, rectifiable \((n-2)\)-varifold with weight measure \(\mu \) (so, in particular, \({\text {spt}}(\mu )\) is \((n-2)\)-rectifiable), and that P(x) is given \(\mu \)-a.e. by the orthogonal projection onto the \((n-2)\)-subspace \(T_x{\text {spt}}(\mu )\subset T_xM\). We collect this information in the following statement.
Proposition 6.4
For a family \((u_{\epsilon },\nabla _{\epsilon })\) satisfying the hypotheses of Theorem 6.1, after passing to a subsequence, there exists a stationary, rectifiable \((n-2)\)-varifold \(V=v(\Sigma ^{n-2},\theta )\) such that
for every continuous section \(S\in C^0(M,{\text {Sym}}(TM))\). The energy measure \(\mu \) is given by 
. Also, we can choose \(\Sigma :={\text {spt}}(\mu )\) and \(\theta (x):=\Theta _{n-2}(\mu ,x)\).
6.2 Integrality of the limit varifold and convergence of level sets
We now show that the varifold V is integer rectifiable. Given \(x\in {\text {spt}}(\mu )\) and \(s>0\), we define \(M_{x,s}\) to be the ball of radius \(s^{-1}{\text {inj}}(M)\) in the Euclidean space \((T_xM,g_x)\) and define \(\iota _{x,s}:M_{x,s}\rightarrow M\) by \(\iota _{x,s}(y):=\exp _x(sy)\). We endow \(M_{x,s}\) with the smooth metric \(g_{x,s}:=s^{-2}\iota _{x,s}^*g\), which converges locally smoothly to the Euclidean metric \(g_x\) as \(s\rightarrow 0\).
By rectifiability, for \(\mu \)-a.e. x the dilated varifolds 
in \(M_{x,s}\) satisfy
as \(s\rightarrow 0\), in duality with \(C_c(T_xM)\). Fix \(x\in {\text {spt}}(\mu )\) such that (6.12) holds. The integrality of V will follow once we prove that \(\Theta =\Theta _{n-2}(\mu ,x)\) is an integer.
We can identify \((T_xM,g_x)\) with \({\mathbb {R}}^n\) by a linear isometry such that \(T_x\Sigma =\{0\}\times {\mathbb {R}}^{n-2}\). We also call \(\mu _{x,s}\) the mass measure of \(V_{x,s}\); equivalently,
With a diagonal selection, changing our sequence \(\epsilon \rightarrow 0\) accordingly, we can find scales \(s_\epsilon \rightarrow 0\) such that we have the convergence of Radon measures
where \(({{\widehat{u}}}_\epsilon ,{{\widehat{\nabla }}}_\epsilon )\) is the pullback of \((u_{s_\epsilon \epsilon },\nabla _{s_\epsilon \epsilon })\) by means of \(\iota _{x,s_{\epsilon }}\), and \({{\widehat{\mu }}}_\epsilon \) is the associated energy measure. Note that \(({{\widehat{u}}}_\epsilon ,{{\widehat{\nabla }}}_\epsilon )\) is stationary for \(E_\epsilon \) in the line bundle \(\iota _{x,s_\epsilon }^*L\), with respect to the base metric \(g_{x,s_\epsilon }\). We introduce the notation
Balls will be denoted by \({{\mathcal {B}}}_r(y)\) or \(B_r^n(y)\), depending on whether they are with respect to \(g_{x,s_\epsilon }\) or \(g_{{\mathbb {R}}^n}\), respectively. The volume |E| of a set E will be always understood with respect to the Euclidean metric.
The next proposition, which exploits quantitatively the monotonicity formula, is similar to an estimate in the proof of [26, Lemma 2.1].
Proposition 6.5
As \(\epsilon \rightarrow 0\) we have
Proof
Let \(C_m\) be the constant in Theorem 4.3. We first note that, given \(y\in \{0\}\times {\mathbb {R}}^{n-2}\),
indeed, for any \(\eta >0\), \(B_{r-\eta }^n(y)\subseteq {{\mathcal {B}}}_r(y)\subseteq B_{r+\eta }^n(y)\) eventually. Setting \(y_\epsilon :=\iota _{x,s_\epsilon }(y)\in M\), we deduce that
Pick \(3\le i\le n\) and fix \(R>0\). Choosing \(y:=-2Re_i\), we can apply (4.12) between the radii \(s_\epsilon R\) and \(3s_\epsilon R\) to obtain that
where \(p_i:=\iota _{x,s_\epsilon }(-2Re_i)\) and \(\nu _{R,i}:={\text {grad}}d_{p_i}\). Now (6.13) and the comparability of \(g_{x,s_\epsilon }\) with \(g_{{\mathbb {R}}^n}\) give
where \({{\widetilde{\nu }}}_{R,i}\) is the gradient of the distance function \(d_{-2Re_i}\), both with respect to the metric \(g_{x,s_\epsilon }\). Since eventually \({{\mathcal {B}}}_{3R}(-2Re_i){\setminus } {{\mathcal {B}}}_{R}(-2Re_i)\) includes \(B_2^2\times B_2^{n-2}\) for R big enough, we get
By monotonicity, as \(\epsilon \rightarrow 0\) we have
The smooth convergence \(g_{x,s_\epsilon }\rightarrow g_{{\mathbb {R}}^n}\) gives \({{\widetilde{\nu }}}_{R,i}(y)\rightarrow Y_{R,i}(y):=\frac{y+2Re_i}{|y+2Re_i|}\) uniformly on \(B_2^2\times B_2^{n-2}\). Hence, the bound (6.15) and (6.14) give
Now \(Y_{R,i}\rightarrow e_i=\partial _i\) as \(R\rightarrow \infty \), and the statement follows from (6.16) and the uniform bound (6.15). \(\square \)
We now state the main technical result of the section, which will be shown later. Fix a cut-off function \(\chi \in C^\infty _c(B_2^2)\) with \(\chi (z)=1\) for \(|z|\le \frac{3}{2}\) and \(0\le \chi \le 1\), and let \({{\widehat{\chi }}}(z,t):=\chi (z)\).
Proposition 6.6
There exists \(F_\epsilon \subseteq B_1^{n-2}\) with \(|F_\epsilon |\ge \frac{1}{4}|B_1^{n-2}|\) such that
Before giving the proof, let us see how this implies the integrality of V.
Proof of Theorem 6.1
As \(\epsilon \rightarrow 0\), we have both (6.17) and
as 
.
In view of (6.15) and (6.19), for any vector field \((Y^3,\dots ,Y^n)\in C^\infty _c(B_2^{n-2},{\mathbb {R}}^{n-2})\) we can integrate (4.4) against \({{\widehat{\chi }}}(\sum _{i=3}^n Y^i\partial _i)\) and obtain, in the Euclidean metric,
for some sequence \(\lambda _\epsilon \rightarrow 0\), thanks to the smooth convergence \(g_{x,s_\epsilon }\rightarrow g_{{\mathbb {R}}^n}\).
Invoking Proposition 6.5 and noting that \(\Vert Y\Vert _{L^\infty }\le 2\Vert DY\Vert _{L^\infty }\), we can conclude that the nonnegative function \(f_\epsilon (t):=\frac{1}{2\pi }\int _{{\mathbb {R}}^2\times \{t\}}{{\widehat{\chi }}} e_\epsilon ({{\widehat{u}}}_\epsilon ,{{\widehat{\nabla }}}_\epsilon )\) satisfies
for a possibly different sequence \(\lambda _\epsilon \rightarrow 0\). Applying the Hahn–Banach theorem to the subspace \(\{DY\mid Y\in C^\infty _c(B_2^{n-2},{\mathbb {R}}^{n-2})\}\subseteq C_0(B_2^{n-2},{\mathbb {R}}^{n-2}\otimes {\mathbb {R}}^{n-2})\) (\(C_0\) denoting the closure of \(C_c\)), we can find real measures \((\nu _\epsilon )^i_j\) such that
as distributions and \(|(\nu _\epsilon )^i_j|(B_2^{n-2})\rightarrow 0\). Allard’s strong constancy lemma [2, Theorem 1.(4)] gives then
Since the sets \(F_{\epsilon }\) of Proposition 6.6 have positive measure, there clearly exists \(t_\epsilon \in F_\epsilon \) such that
Recalling (6.17), we deduce that
Hence, by (6.18), we get \({\text {dist}}(\Theta ,{\mathbb {N}})=0\), which concludes the proof that V is integral. \(\square \)
Proof of Proposition 6.6
Taking into account Proposition 6.5, the classical Hardy–Littlewood weak-(1,1) maximal estimate (applied to the function \(t\mapsto \int _{B_2^2\times \{t\}}e_\epsilon ^T({{\widehat{u}}}_\epsilon ,{{\widehat{\nabla }}}_\epsilon )\)) gives
for all \(t\in B_1^{n-2}{\setminus } E_1^\epsilon \) and \(0<r<1\), where \(E_1^\epsilon \) is a Borel set with \(|E_1^\epsilon |\le \frac{1}{4}|B_1^{n-2}|\). Similarly, (6.15) and (6.19) give
for \(t\in B_1^{n-2}{\setminus }(E_2^\epsilon \cup E_3^\epsilon )\) and \(0<r<1\), with \(|E_2^\epsilon |,|E_3^\epsilon |\le \frac{1}{4}|B_1^{n-2}|\).
Pick any \(t^\epsilon \in B_1^{n-2}{\setminus }(E_1^\epsilon \cup E_2^\epsilon \cup E_3^\epsilon )\) and, for \(0<r<1\), define
(with the Euclidean distance), where \(Z_{\beta _d/2}({{\widehat{u}}}_\epsilon )=\{|{{\widehat{u}}}_\epsilon |^2\le 1-\beta _d/2\}\). In other words, \({\mathcal {V}}^\epsilon \) is the \(t^\epsilon \)-slice of the neighborhood \(B_r^n(Z_{\beta _d/2}({{\widehat{u}}}_\epsilon ))\).
We claim that, for \(0<r<\frac{1}{2}\), \({\mathcal {V}}^{\epsilon }(r)\) satisfies a uniform area bound
provided \(\epsilon <r\) and \(\epsilon \) is small enough. Indeed, \({\mathcal {V}}^\epsilon (r)\times \{t^\epsilon \}\) is covered by the balls \(B_{r}^n(y)\) with \(y\in (B_{3/2}^2\times B_r^{n-2}(t^\epsilon ))\cap Z_{\beta _d/2}({{\widehat{u}}}_\epsilon )\). Vitali’s covering lemma gives a disjoint collection \(\{B_{r}^n(y_j)\mid j\in J\}\) such that \({\mathcal {V}}^\epsilon (r)\times \{t^\epsilon \}\subseteq \bigcup _{j}B_{5r}^n(y_j)\). By Corollary 4.4, we have a bound on the cardinality |J|:
(since \(\frac{1}{4}g_{{\mathbb {R}}^n}\le g_{x,s_\epsilon }\le 4g_{{\mathbb {R}}^n}\) for \(\epsilon \) sufficiently small). Using also (6.21), we get \(|J|\le C(M,\Lambda )\). Hence, writing \(y_j=(z_j,t_j)\), we obtain
confirming (6.23).
Given \(R>0\), let \(\{z_1^\epsilon ,\dots ,z_{N(R,\epsilon )}^\epsilon \}\) be a maximal subset of \({\mathcal {V}}^\epsilon (R\epsilon )\) with \(|z_k^\epsilon -z_\ell ^\epsilon |\ge 2\epsilon \). Since \(\bigcup _k (B_1^2\cap B_\epsilon ^2(z_k))\subseteq {\mathcal {V}}^\epsilon ((R+1)\epsilon )\) and the balls \(B_\epsilon ^2(z_k)\) are disjoint, (6.23) gives a uniform bound on \(N(R,\epsilon )\) independent of \(\epsilon \) (eventually), so up to subsequences we can assume that \(N(R)=N(R,\epsilon )\) is constant and that \(\epsilon ^{-1}|z_k^\epsilon -z_\ell ^\epsilon |\) has a limit \(r_{k\ell }\) as \(\epsilon \rightarrow 0\), for each k, l.
We say that \(k\sim \ell \) if \(r_{k\ell }<\infty \); this is evidently an equivalence relation (as \(r_{km}\le r_{k\ell }+r_{\ell m}\)), so we can pick a set of representatives \(\{k_1,\dots ,k_P\}\) of the distinct equivalence classes \([k_1],\dots ,[k_P]\) and conclude that
eventually, for any fixed \(S\ge S_0(R):=\max \{\sum _{\ell \in [k_j]}r_{k_j\ell }+2\mid j=1,\dots ,P\}\).
Fix such an S which is also bigger than the constants C in (6.21) and \(a_d^{-1},C_d\) in Corollary 5.4. For any fixed \(\delta >0\), (6.20) and (6.21) show that, for \(\epsilon \) sufficiently small, Proposition 6.7 below applies to \({{\widehat{u}}}_\epsilon (z_{k_j}^\epsilon +\epsilon \cdot ,t^\epsilon +\epsilon \cdot )\) (with \(\beta :=\beta _d\)). Writing \(K=K(\beta _d,\delta ,S)>S\), note that the balls \(B_{K\epsilon }^2(z_{k_j})\) are eventually disjoint and included in \(\{\chi =1\}\). Hence, Proposition 6.7 and (6.22) give
(for \(\epsilon \) sufficiently small). Choosing \(\delta =\delta (R)\le \frac{1}{(P+1)R}\), we arrive at the estimate
To conclude the proof, it suffices to show that
Once we have this, we infer that
for the original sequence \((t^\epsilon )\). Noting that the choice of \(t^{\epsilon }\) in \(F_{\epsilon }:=B_1^{n-2}{\setminus } E_1^{\epsilon }\cup E_2^{\epsilon }\cup E_3^{\epsilon }\) was arbitrary, we get
Since the argument applies to an arbitrary subsequence \(\epsilon _j\rightarrow 0\), the proposition then follows.
To show (6.24), note that for \(z\in B_1^2\) the distance of \(\iota _{x,s_\epsilon }((z,t^\epsilon ))\) to the set \(Z_{\beta _d/2}(u_{s_\epsilon \epsilon })\) is (eventually) bounded below by \(\frac{s_\epsilon }{2}\min \{1,r_\epsilon (z)\}\), where \(r_\epsilon (z)\) is the (Euclidean) distance of \((z,t^\epsilon )\) to \(Z_{\beta _d/2}({{\widehat{u}}}_\epsilon ))\). Since \(Z_{\beta _d/2}(u_{s_\epsilon \epsilon })\supseteq Z_{\beta _d}(u_{s_\epsilon \epsilon })\), for any \(R>1\) Corollary 5.4 gives
where we used Fubini’s theorem in the second equality. The statement follows. \(\square \)
The following key technical proposition, used in the proof of Proposition 6.6, relies ultimately on the quantization phenomenon for the energy of entire solutions in the plane, presented in [21, Chapter III]. For the reader’s convenience, we give a self-contained proof, including the relevant arguments from [21].
Proposition 6.7
Given \(0<\beta ,\delta <\frac{1}{2}\) and \(S>1\), there exist \(K(\beta ,\delta ,S)>S\) and \(0<\kappa (\beta ,\delta ,S,n)<K^{-1}\) such that the following is true. Assume \((u,\nabla )\) is smooth and solves (2.4) and (2.5), with \(|u|\le 1\) and \(\epsilon =1\), on a line bundle L over a cylinder (Q, g), with \(Q=B_{\kappa ^{-1}}^2\times B_{\kappa ^{-1}}^{n-2}\). If we have
the energy bounds
as well as the decay
and \(\Vert g-g_{{\mathbb {R}}^n}\Vert _{C^2}\le \kappa \), then
where p is the degree of \(\frac{u}{|u|}(S\cdot ,0)\), as a map from the circle to itself.
Proof
To begin with, fix a real number \(K(\beta ,\delta ,S)>S\) so big that
Arguing by contradiction, assume there exists a sequence \(\kappa _j\rightarrow 0\) such that the statement admits a counterexample \((u_j,\nabla _j)\) (for \(\kappa =\kappa _j\)) for a (necessarily trivial) line bundle \(L_j\) over \(Q_j=B_{\kappa _j^{-1}}^2\times B_{\kappa _j^{-1}}^{n-2}\), with respect to a metric \(g=g_j\) satisfying \(\Vert g-g_{{\mathbb {R}}^n}\Vert _{C^2}\le \kappa _j\). Fixing a trivialization of \(L_j\) over \(Q_j\), we can write \(\nabla _j=d-iA_j\) for some real one-form \(A_j\).
By virtue of the uniform pointwise estimate (6.28) for \(e_1(u_j,\nabla _j)\ge |d|u_j||^2\), we see that the functions \(|u_j|\) are locally equi-Lipschitz. In particular, we can apply the Arzelà–Ascoli theorem to extract a subsequence \(|u_j|\) converging in \(C^0_{loc}\) to a continuous function \(\rho _\infty :{\mathbb {R}}^n\rightarrow {\mathbb {R}}\).
Since \(|\partial _k |u_j||\le |(\nabla _j)_{\partial _k} u_j|\) for all k, (6.27) implies that \(\rho _\infty \) depends only on the first two variables. Moreover, (6.25) gives \(\rho _\infty ^2\ge 1-\frac{\beta }{2}>1-\beta \) outside \(B_S^2\times {\mathbb {R}}^{n-2}\). In particular, setting
we have \(R_j\rightarrow \infty \). Let \(w_j:=\frac{u_j}{|u_j|}\) on \(\{|u_j|>\frac{1}{2}\}\).
The degree \(p_j\) is uniformly bounded as, for \(r\ge S\) and \(t\in {\mathbb {R}}^{n-2}\),
for j sufficiently large, so averaging over \(S<r<2S\) and \(t\in B_1^{n-2}\) we get
as \(|u_j||w_j^*(d\theta )-A_j|\le |\nabla _j u_j|\). Thus, up to subsequences we can assume \(p_j=p\) is constant.
We now claim that, up to change of gauge, \((u_j,A_j)\rightarrow (u_\infty ,A_\infty )\) subsequentially in \(C^{1}_{loc}({\mathbb {R}}^2\times B_1^{n-2})\). Let \({{\widetilde{u}}}_j=e^{i\theta _j}u_j\) and \({{\widetilde{A}}}_j=A_j+d\theta _j\) be the section and the connection in the Coulomb gauge on the domain \(({{\overline{B}}}_{5S}^n,g_j)\), with \({{\widetilde{A}}}_j(\nu )=0\) on the boundary (as described in the “Appendix”). Note that \(B_{5S}^n\) includes the cylinder \(Q':=B_{4S}^2\times B_1^{n-2}\), and observe that, on \(Q'':=(B_{4S}^2{\setminus } B_S^2)\times B_1^{n-2}\), \({\widetilde{u}}_j\) has the form
for a unique real function \(\psi _j\) with \(0\le \psi _j(2S,0)<2\pi \).
Hence, \(u_j=|u_j|e^{i(p\theta +\psi _j-\theta _j)}\) on \(Q''\) and we can extend \(\psi _j-\theta _j\) uniquely to a function \(\sigma _j:(B_{R_j}^2{\setminus } B_S^2)\times B_1^{n-2}\rightarrow {\mathbb {R}}\) so that \(u_j=|u_j|e^{ip\theta +i\sigma _j}\) holds true on all the domain of \(\sigma _j\). Finally, we replace \((u_j,A_j)\) with \((e^{i\tau _j}u_j,A_j+d\tau _j)\), where
for a fixed smooth function \(\chi :[0,\infty )\rightarrow [0,1]\) such that \(\chi =0\) on [0, 2S] and \(\chi =1\) on \([3S,\infty )\). Observe that, in the cylinder \(Q'=B_{4S}^2\times B_1^{n-2}\), the new couple equals
The function \(\psi _j\) obeys uniform local \(W^{2,q}\) bounds, on (the interior of) \(Q''\), for all \(1\le q<\infty \), thanks to the Coulomb gauge specification (per Proposition A.1 in the “Appendix”). Hence, the new couple \((u_j,A_j)\) has uniform local \(W^{2,q}\) bounds on \(Q'\).
Moreover, in the exterior annular region \({\mathcal {A}}_j:=(B_{R_j}^2{\setminus }{{\overline{B}}}_{3S}^2)\times B_1^{n-2}\), we have that \(u_j(re^{i\theta },t)=|u_j|e^{pi\theta }\) and we can obtain local \(W^{2,q}\) bounds noting that
Indeed, since the right-hand side is bounded by \(2e_1(u_j,\nabla _j)^{1/2}\le 2S^{1/2}\) and \(pd\theta \) is a fixed smooth one-form, we immediately obtain uniform \(L^{\infty }\) bounds for \(A_j\) locally in \({\mathcal {A}}_j\). Next, note that the identity (3.4) applies to give us an estimate
in \({\mathcal {A}}_j\), from which it follows that the modulus \(|u_j|\) satisfies uniform \(W^{2,q}\) bounds for every \(q\in (1,\infty )\) locally in \({\mathcal {A}}_j\). Multiplying (2.4) by \(e^{-pi\theta }\) and taking the imaginary part gives
from which it follows that \(d^*A_j\) satisfies uniform \(L^{\infty }\) bounds locally in \({\mathcal {A}}_j\) as well; together with the obvious pointwise bound \(|dA_j|\le e_1(u_j,\nabla _j)^{1/2}\le S^{1/2}\), this in particular yields uniform bounds on the full derivative \(\Vert DA_j\Vert _{L^q}\) for every \(q\in (1,\infty )\) on fixed compact subsets of \({\mathcal {A}}_j\) (this follows, e.g., from [20, Lemma 4.7] and a cut-off argument).
Finally, writing (2.5) as
the preceding chain of identities and estimates give a uniform \(L^q\) bound on the right-hand side over any fixed compact subset of \({\mathcal {A}}_j\), for any \(q\in (1,\infty )\); in particular, this gives us the desired uniform local \(W^{2,q}\) bounds for \(A_j\) (while we already have the desired \(W^{2,q}\) bounds for \(u_j=|u_j|e^{pi\theta }\)).
Thanks to the compact embedding \(W^{2,q}\hookrightarrow C^1\) on bounded regular domains (for \(q>n\)), we obtain a limit couple \((u_\infty ,A_\infty )\) on \({\mathbb {R}}^2\times B_1^{n-2}\), as claimed, which solves (2.4) and (2.5) with respect to the flat metric. Also, \(|u_\infty |=\rho _\infty \) and
The second part of (6.30) implies that we can find a function \(\alpha \in C^1({\mathbb {R}}^2\times B_1^{n-2})\) with \(\alpha (z,0)=0\) and \(\partial _k\alpha =(A_\infty )_k\), for all \(z\in {\mathbb {R}}^2\) and all \(k\ge 3\). Set \({{\widetilde{u}}}_\infty :=e^{-i\alpha }u_\infty \) and \({{\widetilde{A}}}_\infty :=A_\infty -d\alpha \), so that
for all \(k=3,\dots ,n\) and \(\ell =1,\dots ,n\) [using again the second part of (6.30)]. The first part gives instead \(\partial _k{{\widetilde{u}}}_\infty =0\) for \(k=3,\dots ,n\). Hence, \(({{\widetilde{u}}}_\infty ,{{\widetilde{A}}}_\infty )\) depends only on the first two variables and therefore corresponds to a planar solution of (2.4) and (2.5).
Also, from (6.28) we deduce that
for \(|z|>S\), as eventually \({{\overline{B}}}_{|z|-S}^n(z,t)\cap Z_\beta (u_j)=\emptyset \).
Integrating (4.4) on \({\mathbb {R}}^2={\mathbb {R}}^2\times \{0\}\) against the position vector field we get
Thanks to the decay of \(e_1({{\widetilde{u}}}_\infty ,{{\widetilde{A}}}_\infty )\), we can repeat the proof of (3.6): starting from
and applying the maximum principle, we deduce that the decaying function \({{\widetilde{\xi }}}_\infty \) is nonpositive. We then obtain \(|d{{\widetilde{A}}}_\infty |\le \sqrt{W({{\widetilde{u}}}_\infty )}\), so we must have \(|d{{\widetilde{A}}}_\infty |=\sqrt{W({{\widetilde{u}}}_\infty )}\) everywhere (cf. [21, Section III.10]).
Observe that, by (3.4) and the strong maximum principle, \(|{{\widetilde{u}}}_\infty |<1\) (unless \(|{{\widetilde{u}}}_\infty |=1\) everywhere, in which case \(|d{{\widetilde{A}}}_\infty |=\sqrt{W({{\widetilde{u}}}_\infty )}=0\) and \(|{{\widetilde{\nabla }}}_\infty {{\widetilde{u}}}_\infty |=0\) by (3.4), thus \(e_1({{\widetilde{u}}}_\infty ,{{\widetilde{A}}}_\infty )=0\) and \(p=0\); so the statement of the proposition holds eventually, contradiction). As a consequence, \(|*d{{\widetilde{A}}}_\infty |=W({{\widetilde{u}}}_\infty )>0\) and we get either \(\frac{1-|{{\widetilde{u}}}_\infty |^2}{2}=*d{{\widetilde{A}}}_\infty \) everywhere or \(\frac{1-|{{\widetilde{u}}}_\infty |^2}{2}=-*d{{\widetilde{A}}}_\infty \) everywhere. Thus, integrating by parts and using (2.4), as well as the decay of \(|pd\theta -{{\widetilde{A}}}_\infty |\),
Hence, the energy of the two-dimensional solution \(({{\widetilde{u}}}_\infty ,{{\widetilde{A}}}_\infty )\) is \(2\pi |p|\). Our choice of K, namely (6.29), together with (6.31), then ensures that
As a consequence, this must hold eventually also for \((u_j,A_j)\), giving the desired contradiction. \(\square \)
Remark 6.8
As a consequence, one also finds that
if \(|u|>0\) everywhere on the cylinder Q. Indeed, if \(|u|>0\) everywhere, then the degree p in the statement of Proposition 6.7 clearly must vanish.
We are now able to address the statement on the convergence of level sets.
Proposition 6.9
For any \(0\le \delta <1\) we have \({\text {spt}}(\mu )=\lim _{\epsilon \rightarrow 0}\{|u_\epsilon |\le \delta \}\), in the Hausdorff topology.
Proof
If \(x=\lim _{\epsilon \rightarrow 0}x_\epsilon \), for points \(x_\epsilon \in \{|u_\epsilon |\le \delta \}\) defined along a subsequence, then the same argument used in the proof of Proposition 6.2 shows that \(x\in {\text {spt}}(\mu )\). Hence, for all \(\eta >0\), eventually \(\{|u_\epsilon |\le \delta \}\) is included in the \(\eta \)-neighborhood of \({\text {spt}}(\mu )\).
To conclude the proof, it suffices to show that the converse inclusion \({\text {spt}}(\mu )\subseteq B_\eta (\{u_\epsilon =0\})\) holds eventually. Arguing by contradiction, assume that there are points \(p_\epsilon \in {\text {spt}}(\mu )\) whose distance from \(\{u_\epsilon =0\}\) is at least \(\eta \), along some subsequence (not relabeled). Up to further subsequences, let \(p_\epsilon \rightarrow p_0\in {\text {spt}}(\mu )\).
Since \(\mu \) is \((n-2)\)-rectifiable, there exists a point \(q\in {\text {spt}}(\mu )\) with \({\text {dist}}(p_0,q)<\frac{\eta }{2}\), and such that \(\mu \) blows up to 
at q. Observe that eventually we have
Now, repeating all the preceding blow-up analysis at q, in view of Remark 6.8 we can improve (6.17) to the uniform convergence
for \(t\in F_\epsilon \), which implies that \(\Theta _{n-2}(\mu ,q)=0\). However, since \(q\in {\text {spt}}(\mu )\), this is impossible, by Proposition 6.2. \(\square \)
6.3 Limiting behavior of the curvature
As before, we identify the curvature \(F_{\nabla _\epsilon }\) with a closed two-form \(\omega _{\epsilon }\) by \(F_{\nabla _{\epsilon }}(X,Y)=-i\omega _{\epsilon }(X,Y)\). Recall that the cohomology class \([\frac{1}{2\pi }\omega _{\epsilon }]\) represents the (rational) first Chern class \(c_1(L)\in H^2(M;{\mathbb {R}})\) of the complex line bundle \(L\rightarrow M\).
Theorem 6.10
Let \((u_{\epsilon },\nabla _{\epsilon })\) be a family as in Theorem 6.1. The curvature forms \(\frac{1}{2\pi }\omega _{\epsilon }\) can be identified with \((n-2)\)-currents that converge (weakly), as \(\epsilon \rightarrow 0\), to an integer rectifiable cycle \(\Gamma \) which is Poincaré dual to \(c_1(L)\), and whose mass measure \(|\Gamma |\) satisfies \(|\Gamma |\le \mu .\)
Proof
Recall from Sect. 2 that
where \(\psi (u_{\epsilon })=\langle 2i\nabla u_{\epsilon },\nabla _{\epsilon } u_{\epsilon }\rangle \) is a two-form satisfying \(|\psi (u_\epsilon )|\le |\nabla _\epsilon u_\epsilon |^2\) pointwise. In particular, denoting by \(J(u_{\epsilon },\nabla _{\epsilon })\) the two-form
we can rewrite this identity as
and observe that
The dual \((n-2)\)-currents given by
for any \((n-2)\)-form \(\zeta \in \Omega ^{n-2}(M)\), are thus bounded in mass by \(\frac{1}{2\pi }\Lambda \). (Here we compute the mass with the \(\ell ^2\) norm on exterior algebras; for the limit current, by rectifiability this will coincide with the usual mass, dual to the comass.) Up to subsequences, we can take a weak limit \(\Gamma \). The bound \(|\Gamma _\epsilon |\le \mu _{\epsilon }\) implies that also \(|\Gamma |\le \mu \).
From (6.33) and integration by parts we get
Since (as discussed in the proof of Proposition 6.2)
as \(\epsilon \rightarrow 0\), it follows that
for every smooth \((n-2)\)-form \(\zeta \in \Omega ^{n-2}(M)\).
Since the two-forms \(\omega _{\epsilon }\) are closed, for any \(\xi \in \Omega ^{n-3}(M)\) we have
so \(\Gamma \) is a cycle. Since \(\mu \) is \((n-2)\)-rectifiable, \(\Gamma \) must be a rectifiable \((n-2)\)-current: this can be seen by blow-up, applying [25, Proposition 7.3.5]. By (6.35), \(\Gamma \) is Poincaré dual to \(c_1(L)\).
To complete the proof, it remains to show that \(\Gamma \) has integer multiplicity. By means of a diagonal selection of a subsequence, as in the previous subsection, we can deduce integrality at those points \(p\in {\text {spt}}(\mu )\) where \(\mu \) and \(\Gamma \) blow up respectively to 
and a multiple of \([T_p\Sigma ]\), using the following lemma. Note that its hypotheses are verified thanks to Corollary 5.4 and the fact that \(Z_{\beta _d}(u_\epsilon )\) necessarily converges to a subset of \(T_p\Sigma \) in the local Hausdorff topology, after rescaling (see the proof of Proposition 6.2).
Since \(\mu \) is \((n-2)\)-rectifiable, we deduce that the limiting current \(\Gamma \) has integer multiplicity \({\mathcal {H}}^{n-2}\)-a.e. on its support, as claimed. \(\square \)
Lemma 6.11
On the Euclidean ball \(B_4^n\), let \((u_{\epsilon },\nabla _{\epsilon })\) be a sequence of sections and connections in a trivial line bundle \(L\rightarrow B_4^n\) (not necessarily satisfying any equation) for which \(E_{\epsilon }(u_{\epsilon },\nabla _{\epsilon })\le \Lambda \), \(e_{\epsilon }(u_{\epsilon },\nabla _{\epsilon })\rightarrow 0\) in \(C^0_{loc}(B_4^n{\setminus } P)\) and \(*\omega _{\epsilon }\rightarrow \theta _1 [P]\) in \({\mathcal {D}}_{n-2}(B_4^n)\), where \(P=\{0\}\times {\mathbb {R}}^{n-2}\). Then \(\theta _1\in 2\pi {\mathbb {Z}}\).
Proof
To begin, fix a test function \(\varphi \in C_c^1(B_1^2\times B_1^{n-2})\) of the form \(\varphi (x^1,\ldots ,x^n)=\psi (x^1,x^2)\eta (x^3,\ldots ,x^n)\), with \(\psi (x^1,x^2)=1\) for \(|(x^1,x^2)|\le \frac{1}{2}\). In the sequel, we shall omit the domain of integration when it equals \({\mathbb {R}}^n\). By assumption, we then have
Fixing trivializations of L over \(B_2^n\), we write \(\nabla _{\epsilon }=d-iA_{\epsilon }\) for some one-forms \(A_{\epsilon }\), so that \(\omega _{\epsilon }=dA_{\epsilon }\), and the right-hand term in the preceding limit becomes
On \(B_2^n\) we can choose our trivializations so that \(d^*A_{\epsilon }=0\), and \(A_{\epsilon }(\nu )=0\) on \(\partial B_2^n\) (see the “Appendix”). We then have the \(L^2\) control
(see, e.g., [20, Theorem 4.8]), and consequently
as \(\epsilon \rightarrow 0\), where we have used the fact that \(d\psi (x^1,x^2) = 0\) for \(|(x^1,x^2)|\le \frac{1}{2}\), and the assumption that \(e_{\epsilon }(u_{\epsilon },\nabla _{\epsilon })\rightarrow 0\) in \(C^0_{loc}(B_2^n{\setminus } P)\).
Combining our computations thus far, we have arrived at the identity
Noting next that
and using again the hypothesis that \(e_{\epsilon }(u_{\epsilon },\nabla _{\epsilon })\rightarrow 0\) uniformly on \({\text {spt}}(\eta d\psi )\), the preceding identity yields
Finally, since the one-form \((u_{\epsilon }/|u_{\epsilon }|)^*(d\theta )\) is closed on \(\{u_\epsilon \ne 0\}\) and \(d\eta \wedge dx^3\wedge \dots \wedge dx^n=0\), integrating by parts on \(({\mathbb {R}}^2{\setminus } B_{1/2}^2)\times {\mathbb {R}}^{n-2}\) we see that
where \({\text {deg}}(u_\epsilon ,P)\) stands for the degree of \((u_\epsilon /|u_\epsilon |)(\frac{1}{2}e^{i\theta },0)\). The statement follows. \(\square \)
7 Examples from variational constructions
The goal of this section is to show that, for every closed manifold M and every line bundle \(L\rightarrow M\) endowed with a Hermitian metric, there exist critical couples \((u_{\epsilon },\nabla _{\epsilon })\) for the Yang–Mills–Higgs functional \(E_\epsilon \), for \(\epsilon \) small enough, in such a way that
This will be easier when the line bundle is nontrivial, as in this case we can just take \((u_\epsilon ,\nabla _\epsilon )\) to be a global minimizer for \(E_\epsilon \). The upper and lower bounds in (7.1) have the following immediate consequence—proved previously by Almgren [5] using GMT methods.
Corollary 7.1
Every closed Riemannian manifold \((M^n,g)\) supports a nontrivial stationary, integral \((n-2)\)-varifold.
Proof
We can always equip M with the trivial line bundle \(L:={\mathbb {C}}\times M\). As shown in the next subsection, there exists a sequence of critical couples \((u_{\epsilon },\nabla _{\epsilon })\) satisfying (7.1). The statement now follows from Theorem 6.1. \(\square \)
7.1 Min–max families for the trivial line bundle
In this section we will show how min–max methods may be applied to the functionals \(E_{\epsilon }\) to produce nontrivial critical points in the trivial bundle \(L={\mathbb {C}}\times M\) on an arbitrary closed manifold M of dimension \(n\ge 2\). The min–max construction that we consider here is based on two-parameter families parametrized by the unit disk, similar to the constructions employed in [10, 33] for the Ginzburg–Landau functionals—with several technical adjustments to account for the gauge-invariance and other features particular to the Yang–Mills–Higgs energies.
One can show that the families we consider induce a nontrivial class in \(\pi _2({\mathcal {M}})\) for the quotient
and the analysis that follows can be reformulated in terms of min–max methods applied directly to \({\mathcal {M}}\), which can be given the structure of a Banach manifold.
Without loss of generality, we assume henceforth that M is connected. In some proofs we will also implicitly assume that \(n={\text {dim}}(M)\ge 3\), leaving the obvious changes for \(n=2\) to the reader.
Definition 7.2
Fix \(n=\dim (M)<p<\infty \). In what follows, \({{\widehat{X}}}\) will denote the Banach space of couples (u, A), where \(u\in L^p(M,{\mathbb {C}})\) and \(A\in \Omega ^1(M,{\mathbb {R}})\), both of class \(W^{1,2}\), with the norm
Denote by \(X:=\{(u,A)\in {\widehat{X}}:d^*A=0\}\) the subspace consisting of those couples for which the connection form A is co-closed.
Note that, for \((u,A)\in X\), the full covariant derivative \(\int _M|DA|^2\) is bounded by \(C(M)\int _M(|A|^2+|dA|^2)\): see, e.g., [20, Theorem 4.8] for a proof.
Definition 7.3
Given a form \(A\in \Omega ^1(M,{\mathbb {R}})\) in \(L^2\), we denote by h(A) the harmonic part of its Hodge decomposition, or equivalently the orthogonal projection of A onto the (finite-dimensional) space \({\mathcal {H}}^1(M)\) of harmonic one-forms.
Remark 7.4
Selection of a Coulomb gauge gives a continuous retraction \({\mathcal {R}}:{{\widehat{X}}}\rightarrow X\): namely, given a couple \((u,A)\in {{\widehat{X}}}\), consider the unique solution \(\theta \in W^{2,2}(M,{\mathbb {R}})\) to the equation
with \(\int _M\theta =0\), and set
Note that the continuity of \((u,A)\mapsto d(e^{i\theta }u)=e^{i\theta }(du+iud\theta )\), from \({\widehat{X}}\) to \(L^2\), follows from the fact that \(L^p\cdot L^{2^*}\subseteq L^2\), where \(2^*=\frac{2n}{n-2}\).
Throughout this section, \(W(u)=f(|u|)\) will be a smooth radial function given by \(W(u)=\frac{(1-|u|^2)^2}{4}\) for \(|u|\le 3/2\), and satisfying \(W(u),W'(u)[u]>0\) for all \(|u|>1\). For technical reasons, we also find it convenient to require that
which evidently gives the additional estimates \(|u|f'(|u|)+|u|^2f''(|u|)\le C|u|^p\) for \(|u|\ge 2\), for some constant C. For future use, observe also that the potential W(u) then satisfies a simple bound of the form
Proposition 7.5
The functional \(E_\epsilon \) is of class \(C^1\) on \({\widehat{X}}\). Moreover, a couple (u, A) is critical in \({{\widehat{X}}}\) for \(E_\epsilon \) if and only if \({\mathcal {R}}((u,A))\) is critical in X. Critical points are smooth up to change of gauge.
Proof
Given a point \((u,A)\in {{\widehat{X}}}\) and a pair \((v,B)\in {{\widehat{X}}}\) with \(\Vert (v,B)\Vert _{{{\widehat{X}}}}\le 1\), direct computation gives
where we are using the fact that \({{\widehat{X}}} \cdot {{\widehat{X}}}\subseteq L^n\cdot L^{2^*}\subseteq L^2\) to see that
and we invoke our assumptions on the structure of W to see that
for fixed \((u,A)\in {{\widehat{X}}}\). It follows immediately that \(E_{\epsilon }\) is \(C^1\) on \({{\widehat{X}}}\), with differential
To confirm the second statement, assume without loss of generality that v and B are smooth, and observe that
where \(({{\widetilde{u}}},{{\widetilde{A}}}):={\mathcal {R}}((u,A))=(e^{i\theta }u,A+d\theta )\) and \(\psi \) solves \(\Delta \psi =d^*B\). This easily gives
and, using the gauge invariance \(E_\epsilon =E_\epsilon \circ {\mathcal {R}}\), we deduce that
It follows that if \(({{\widetilde{u}}},{{\widetilde{A}}})\) is critical for \(E_{\epsilon }\) in X then (u, A) is critical for \(E_{\epsilon }\) in \({{\widehat{X}}}\), as claimed. The converse is similar.
Finally, if (u, A) is critical for \(E_\epsilon \) (in either \({\widehat{X}}\) or X), then applying the above formula for the differential with \(v=(|u|-1)^+u/|u|\in W^{1,2}\) and \(B=0\) we get
where we used the fact that \(\langle u\otimes d((|u|-1)^+/|u|),\nabla u\rangle \) equals \(|u|^{-1}|d|u||^2\ge 0\) a.e. on \(\{|u|>1\}\) and vanishes elsewhere. Since \(W'(u)[u]>0\) on \(\{|u|>1\}\) by our assumption on W, we deduce that \(|u|\le 1\). Together with Proposition A.1 and Remark A.3 in the “Appendix”, this implies that (u, A) is smooth in an appropriate (Coulomb) gauge. \(\square \)
We next show that the functionals \(E_{\epsilon }\) satisfy a suitable variant of the Palais–Smale condition on X, giving compactness of critical sequences for \(E_{\epsilon }\) after an appropriate change of gauge. (Cf. [23] for similar results in the Seiberg–Witten setting.)
Proposition 7.6
The functional \(E_\epsilon \) satisfies the following form of the Palais–Smale condition: every sequence \((u_j,A_j)\) in X with bounded energy and \(dE_\epsilon (u_j,A_j)\rightarrow 0\) in \(X^*\) admits a subsequence converging strongly in X to a critical couple \((u_\infty ,A_\infty )\), up to possibly replacing \((u_j,A_j)\) with
for suitable smooth harmonic functions \(v_j:M\rightarrow S^1\).
Proof
First, we show that the boundedness of \(E_\epsilon (u_j,A_j)\) implies the boundedness of the sequence in X, up to a change of gauge as in the statement. The assumption (G) on the potential W gives
that is, \(u_j\) is uniformly bounded in \(L^p\).
Denote by \(\Lambda \subseteq {\mathcal {H}}^1(M)\) the lattice in the space of harmonic one-forms given by
and let \(\lambda _j\in \Lambda \) be a closest integral harmonic one-form to \(h(A_j)\) (with respect to the \(L^2\) norm, say, on \({\mathcal {H}}^1(M)\)). Then \(\lambda _j=-v_j^*(d\theta )\) for a suitable harmonic map \(v_j:M\rightarrow S^1\), and
Replacing \((u_j,A_j)\) with the change of gauge \((v_ju_j,A_j-\lambda _j)\in X\), we can then assume that \(h(A_j)\) is bounded.
By standard Hodge theory we can write
for some closed \(\xi _j\in W^{2,2}\) satisfying \(\Delta _H\xi _j=dA_j\) and \(\Vert d^*\xi _j\Vert _{W^{1,2}}\le C(M)\Vert dA_j\Vert _{L^2}\). Thus, given the energy bound \(E_{\epsilon }(u_j,A_j)\le C\), we see that
whereby \(A_j\) is bounded in \(W^{1,2}\) and, consequently, in \(L^{2^*}\). As a consequence, we see next that
taking into account (7.4), we infer then that \(\Vert du_j\Vert _{L^2}\) is also bounded as \(j\rightarrow \infty \).
We have therefore shown that \((u_j,A_j)\) is uniformly bounded in X as \(j\rightarrow \infty \), so passing to subsequences we can assume that \((u_j,A_j)\) converges pointwise a.e. and weakly (in X) to a limiting couple \((u_\infty ,A_\infty )\).
In particular, defining r by
where \(n<q<p\) is an arbitrary fixed exponent, it follows from the compactness of the embedding \(W^{1,2}\hookrightarrow L^{r}\) that
Moreover, the boundedness of \(u_j\) in \(L^p\) and the pointwise convergence to \(u_\infty \) give
By definition of r, this implies in particular that
Next, compute
and observe that, due to the \(L^2\) convergence \(u_jA_k\rightarrow u_{\infty }A_{\infty }\), the right-hand side equals
as \(j,k\rightarrow \infty \). For the difference
we then see that
as \(j,k\rightarrow \infty \).
Now, by our assumption (G) on the structure of W(u), it is not difficult to check (see, e.g., [17, Corollary 1]) that the zeroth order term in our computation for \(D_{j,k}\) satisfies a lower bound
for some constant \(C>0\). In particular, it follows now from the preceding computations and the \(L^1\) convergence \(u_j\rightarrow u_{\infty }\) that
as \(j,k\rightarrow \infty \). On the other hand, since \(d E_{\epsilon }(u_j,A_j)\rightarrow 0\) and \((u_j-u_k,A_j-A_k)\) is bounded in X, we know also that
and it then follows that \((u_j,A_j)\) is Cauchy in X. In particular, \((u_j,A_j)\) converges strongly to \((u_{\infty },A_{\infty })\), which necessarily satisfies
\(\square \)
Having confirmed that the energies \(E_{\epsilon }\) satisfy a Palais–Smale condition, we now argue in roughly the same spirit as [10, 33] to produce nontrivial critical points via min–max methods. To begin, note that the space X splits as \({\mathbb {C}}\oplus Y\), where \({\mathbb {C}}\) is identified with the set of constant couples \((\alpha ,0)\) and
Definition 7.7
Let \(\Gamma \) denote the set of continuous families of couples \(F:{{\overline{D}}}\rightarrow X\) parametrized by the closed unit disk \({{\overline{D}}}\), with
for all \(\theta \in {\mathbb {R}}\). Equivalently, under the above identification \({\mathbb {C}}\subset X\), we require \(F|_{\partial D}={\text {id}}\). We denote by \(\omega _\epsilon (M)\) the “width” of \(\Gamma \) with respect to the energy \(E_\epsilon \), namely
Thanks to Proposition 7.6, we can apply classical min–max theory for \(C^1\) functionals on Banach spaces (see e.g. [15, Theorem 3.2]) to conclude that \(\omega _\epsilon \) is achieved as the energy of a smooth critical couple \((u_\epsilon ,A_\epsilon )\). In the following proposition, we show that \(\omega _{\epsilon }(M)\) is positive, so that the corresponding critical couples \((u_{\epsilon },A_{\epsilon })\) are nontrivial.
Proposition 7.8
We have \(\omega _\epsilon (M)>0\).
Proof
We argue by contradiction, though the proof could be made quantitative. Since we are proving only the positivity \(\omega _{\epsilon }(M)>0\) at this stage—making no reference to the dependence on \(\epsilon \)—in what follows we take \(\epsilon =1\) for convenience. Assume that we have a family \(F\in \Gamma \) with \(\max _{y\in {{\overline{D}}}}E(F(y))<\delta \), with \(\delta \) very small. Writing \(F(y)=(u,A)\), this implies that
When \(b_1(M)\ne 0\), some additional work is required to deduce that the harmonic part h(A) of A must also be small for all couples \((u,A)=F(y)\) in the family. In particular, we will need to employ the following lemma, showing that h(A) lies close to the integral lattice \(\Lambda \subset {\mathcal {H}}^1(M)\) when \(E(u,A)<\delta \).
Lemma 7.9
There exists \(C(M)<\infty \) such that if \((u,A)\in X\) satisfies \(E(u,A)<\delta \), with \(\delta \) small enough, then
Proof
As in [33], it is convenient to define a box-type norm \(|\cdot |_b\) on the space \({\mathcal {H}}^1(M)\) of harmonic one-forms as follows. Fix a collection \(\gamma _1,\ldots ,\gamma _{b_1(M)}\in C^{\infty }(S^1,M)\) of embedded loops generating \(H_1(M;{\mathbb {Q}})\) and, for \(h\in {\mathcal {H}}^1(M)\), set
Since \({\mathcal {H}}^1(M)\) is finite-dimensional, this is of course equivalent to any other norm on \({\mathcal {H}}^1(M)\). Assuming for simplicity that M is orientable, we may fix a collection of diffeomorphisms \(\Phi _i:B_1^{n-1}(0)\times S^1\rightarrow T(\gamma _i)\) onto tubular neighborhoods \(T(\gamma _i)\) of \(\gamma _i\), such that \(\Phi _i(0,\theta )=\gamma _i(\theta )\). For every \(t\in B_1^{n-1}\), set \(\gamma _i^t(\theta ):=\Phi _i(t,\theta )\).
Suppose now that \((u,A)\in X\) satisfies the energy bound
As a consequence of the curvature bound \(\Vert dA\Vert _{L^2}\le \delta ^{1/2}\) and the definition of X, it follows that
as well. As in the proof of Proposition 7.6, applying a gauge transformation \(\phi \cdot (u,A)\) by an appropriate choice of harmonic map \(\phi :M\rightarrow S^1\), we may assume moreover that
which together with the energy bound (7.8) and the definition of X leads us to the estimate
(Note that making a harmonic change of gauge preserves not only the energy E(u, A), but also the distance \({\text {dist}}_b(h(A),\Lambda )\), so it indeed suffices to establish the desired estimate in this gauge.)
Combining these estimates with a simple Fubini argument, we see that there exists a nonempty set S of \(t\in B_1^{n-1}\) for which
and
Recalling the pointwise bound (7.2) for W(u), observe next that
so that, along a curve \(\gamma _i^t\) satisfying (7.10), it follows that
Now, choose \(\delta <\delta _1(M)\) sufficiently small that (7.13) gives
on \(\gamma _i^t\), so that \(\phi :=u/|u|\) defines there an \(S^1\)-valued map \(\phi :\gamma _i^t\rightarrow S^1\), whose degree is given by
When (7.10)–(7.12) hold, we observe next that
Since \(|u|\ge \frac{1}{2}\) on \(\gamma _i^t\), it follows that
as well. Combining this with (7.11), we then deduce that
On the other hand, we already made a gauge transformation so that
So, for \(\delta \) chosen sufficiently small that \(C\delta ^{1/2}<\pi \), it follows that the degree \({\text {deg}}(\phi )=0\). In particular, we can now conclude that
giving the desired estimate. \(\square \)
Remark 7.10
If M is not orientable, we have the weaker conclusion \({\text {dist}}(h(A),\frac{1}{2}\Lambda )\le C\delta ^{1/2}\) (still sufficient for the sequel): indeed, whenever \(\gamma _i\) reverses the orientation, we can still parametrize a double cover of \(T(\gamma _i)\) in the same way, with \(\gamma _i^t\) homotopic to \(\gamma _i\) traveled twice; in this case, the bound (7.15) implies that \(2\int _{\gamma _i}h(A)=\int _{\gamma _i^t}h(A)\) has distance to \(2\pi {\mathbb {Z}}\) bounded by \(C\delta ^{1/2}\), from which the claim follows.
Returning to the proof of Proposition 7.8, suppose again that we have a family \({{\overline{D}}}\ni y \mapsto F(y)\in X\) in \(\Gamma \) with
For \(\delta <\delta _1(M)\) sufficiently small, it follows from the lemma that \({\text {dist}}_b(h(A),\Lambda )<\pi \) for every couple \((u,A)=F(y)\) in the family. In particular, since the assignment \((u,A)\mapsto h(A)\) gives a continuous map \(X\rightarrow {\mathcal {H}}^1(M)\), and since \(h(A)=A=0\) for \(y\in \partial {{\overline{D}}}\), it follows that 0 is the nearest point in the lattice \(\Lambda \) to h(A) for every \(y\in {{\overline{D}}}\), and the estimate therefore becomes
In particular, combining this with (7.6), we see now that
for every couple \((u,A)=F(y)\) in the family.
Now, for \((u,A)=F(y)\), our structural assumption (G) on W(u) gives
which together with the smallness
of A in \(L^{2^*}\) (recalling that \(p>n\)) gives
Combining this with the fact that \(\int _M |du-iuA |^2\le E(u,A)<\delta \) by assumption, we then deduce that
as well.
Finally, by (7.2) and the Poincaré inequality, we have
As a consequence, we find that \(\int _M u_y\) is nonzero for all \((u_y,A_y)=F(y)\) in the family. But then the averaging map
gives a retraction \({{\overline{D}}}\rightarrow \partial {{\overline{D}}}\), whose nonexistence is well known. This gives the desired contradiction. \(\square \)
Having shown positivity \(\omega _{\epsilon }(M)>0\) of the min–max energies, we can now deduce the lower bound in (7.1) from the following simple fact.
Proposition 7.11
There exist \(c(M)>0\) and \(\epsilon _0(M)>0\) such that the following holds, for \(\epsilon \le \epsilon _0\). If \((u,\nabla )\) is critical for the functional \(E_\epsilon \), then either \(E_\epsilon (u,\nabla )\ge c\) or \(E_\epsilon (u,\nabla )=0\).
Remark 7.12
For future reference, we make the obvious observation that the trivial case \(E_{\epsilon }(u,\nabla )=0\) can only occur when the bundle L is trivial.
Proof
By Proposition 7.5, critical points are smooth up to change of gauge. We claim that, whenever \(E_\epsilon (u,\nabla )>0\), u has to vanish at some point \(x_0\in M\). Once we have this, assume e.g. \(E_\epsilon (u,\nabla )\le 1\); Corollary 4.4 (with \(\Lambda =1\)) gives a constant \(\epsilon _0>0\) such that \(r^{2-n}E_\epsilon (u,\nabla ,B_r(x_0))\) has a lower bound independent of \(\epsilon \) and r, for any radius \(\epsilon<r<{\text {inj}}(M)\), provided that \(\epsilon \le \epsilon _0\).
We show the contrapositive, namely we assume that u is nowhere vanishing and show that the energy is zero. Note that L must be trivial and we can use the section \(\frac{u}{|u|}\) to identify L isometrically with the trivial line bundle \({\mathbb {C}}\times M\), equipped with the canonical Hermitian metric. Under this identification, \(u:M\rightarrow {\mathbb {C}}\) takes values into positive real numbers. Writing \(\nabla =d-iA\) and observing that \(\langle \nabla u,iu\rangle =-|u|^2A\), (2.5) becomes
Integrating against A we get \(\int _M(\epsilon ^2|dA|^2+u^2|A|^2)=0\), so \(A=0\) and \(\nabla \) is the trivial connection. At a minimum point \(y_0\) for u, (3.4) gives
which forces \(u(y_0)\ge 1\) and thus \(u=1\) everywhere, giving \(E_\epsilon (u,\nabla )=0\). \(\square \)
Finally, we turn to the uniform upper bound. In the next statement, \(L\rightarrow M\) is a Hermitian line bundle with a fixed Hermitian reference connection \(\nabla _0\). We identify any other Hermitian connection \(\nabla \) with the real one-form A such that \(\nabla s=\nabla _0 s-is\otimes A\) for all sections s.
Proposition 7.13
Given a smooth section \(u:M\rightarrow L\), we can find a smooth couple \((u',A')\) such that
for a universal constant C.
Proof
On \(\{u\ne 0\}\) we let
Note that the compatibility of \(\nabla _0\) with the Hermitian metric on L forces \(\langle \nabla _0w,w\rangle =0\), so that A is a real one-form.
We fix a smooth function \(\rho :[0,\infty ]\rightarrow [0,1]\) with
and we set
where the right-hand side is meant to be zero on \(\{u=0\}\).
Writing \(F_{\nabla _0}=-i\omega _0\), observe that \((\nabla _0-iA)w=0\), hence
In particular, \(e_\epsilon (u',A')=0\) on \(\{|u|>\frac{1}{2}\}\).
From the estimates \(|d|u||\le |\nabla _0 u|\) and \(|A|=|\nabla _0w|\le 2|u|^{-1}|\nabla _0 u|\), it follows that also
and the statement follows immediately. \(\square \)
Proof of (7.1)
The method used in [33, Section 3] gives a continuous map \(H:{{\overline{D}}}\rightarrow W^{1,2}\cap C^0(M,{\mathbb {C}})\) such that \(H(y)\equiv y\) for \(y\in \partial D\) and
for all \(y\in {{\overline{D}}}\)—the full Dirichlet energy having a worse bound \(\int _M|dH(y)|^2\le C\log \epsilon ^{-1}\), which is the natural one in the setting of Ginzburg–Landau. By approximation, we can assume that H takes values in \(C^\infty (M,{\mathbb {C}})\), continuously in y, and still satisfies the same uniform bounds (7.19) (possibly increasing C and replacing \(\frac{3}{4}\) with \(\frac{1}{2}\)).
To each section H(y) of the trivial line bundle, Proposition 7.13 assigns in a continuous way an element \(F(y)\in X\). From the way F(y) is constructed, it is clear that \(F\in \Gamma \). Finally, combining (7.18) with (7.19) gives
\(\square \)
7.2 Minimizers for nontrivial line bundles
Suppose now that L is a nontrivial line bundle, equipped with a Hermitian metric. Fix a smooth Hermitian connection \(\nabla _0\) and identify any other Hermitian connection \(\nabla \) with the real one-form A such that
We can define \({{\widehat{X}}}\) and X as in the previous subsection. With this notation, observe that the curvature of \(\nabla \) is given by
Hence, writing \(F_{\nabla _0}=-i\omega _0\), we have
Definition 7.14
For a fixed \(n<p<\infty \), we define \({{\widehat{X}}}\) to be the Banach space of couples (u, A), where \(u:M\rightarrow L\) is an \(L^p\) section and \(A\in \Omega ^1(M,{\mathbb {R}})\), both of class \(W^{1,2}\), with the norm
We let \(X:=\{(u,A)\in {\widehat{X}}:d^*A=0\}\).
The analogous statements to Remark 7.4 and Propositions 7.5 and 7.6 hold, with identical proofs (replacing du and uA with \(\nabla _0 u\) and \(u\otimes A\), respectively).
Arguing as in the proof of Proposition 7.6, it is easy to see that a minimizing sequence for \(E_{\epsilon }\) in X converges weakly—up to change of gauge—to a global minimizer \((u_{\epsilon },A_{\epsilon })\). We now show that the energy of these minimizers enjoys uniform upper and lower bounds as \(\epsilon \rightarrow 0\).
Proof of (7.1)
The lower bound in (7.1) follows directly from Proposition 7.11 and Remark 7.12. In order to obtain the upper bound, pick a smooth section \(s:M\rightarrow L\) transverse to the zero section (see, e.g., [24, Theorem IV.2.1]) and let \(N:=\{s=0\}\), which is a smooth embedded \((n-2)\)-submanifold of M. Proposition 7.13 applied to \(\epsilon ^{-1}s\) gives a couple \((u_\epsilon ',A_\epsilon ')\) with
By transversality of s, the set \(\{|s|\le \frac{\epsilon }{2}\}\) is contained in a \(C(s)\epsilon \)-neighborhood of N, whose volume is bounded by \(C(s)\epsilon ^2\). We infer that
\(\square \)
Remark 7.15
When M is oriented, N can be oriented in such a way that \([N]\in H_{n-2}(M,{\mathbb {R}})\) is Poincaré dual to the Euler class \(e(L)\in H^2(M,{\mathbb {R}})\) of the line bundle, which equals the first Chern class \(c_1(L)\). The fact that the energy of our competitors concentrates along N suggests that, given a sequence of global minimizers \((u_\epsilon ,A_\epsilon )\), up to subsequences the corresponding energy concentration varifold is induced by an integral mass-minimizing current whose homology class is Poincaré dual to \(c_1(L)\). Theorem 6.10 provides the natural candidate \(\Gamma \), which also satisfies \(|\Gamma |\le \mu \).
Notes
Here and elsewhere, we implicitly identify \(F_{\nabla }\) with the two-form \(\omega \) given by \(F_{\nabla }(X,Y)=-i\omega (X,Y)\).
The precise form of the energies considered by Bradlow in [9] differs slightly from the functionals \(E_{\epsilon }\) considered here, but the analysis is essentially the same.
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Acknowledgements
A hearty thank-you goes to Tristan Rivière for introducing the authors to each other, and for suggesting the line of investigation taken up in the present paper. D.S. also thanks Fernando Codá Marques for his interest in this work, and Francesco Lin for pointing him to the reference [39]. A.P. is partially supported by SNSF Grant 172707. During the completion of this work, D.S. was supported in part by NSF Grant DMS-1502424.
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Appendix: Interior regularity in the Coulomb gauge
Appendix: Interior regularity in the Coulomb gauge
In this short appendix, we describe the essential ingredients needed to establish local regularity in the Coulomb gauge for finite-energy critical points (u, A) of the (\(\epsilon =1\)) abelian Higgs energy E(u, A), collecting some estimates which will be of use elsewhere in the paper.
Consider the manifold with boundary \(({{\overline{\Omega }}}^n,g)\) given by a smooth, contractible domain \(\Omega ^n\subset \subset {\mathbb {R}}^n\) equipped with a \(C^2\) metric g, and let \(L\cong {\mathbb {C}}\times \Omega \) be the trivial line bundle over \(\Omega \), with the standard Hermitian structure. With respect to the metric g, we then define the Yang–Mills–Higgs energy
as in the preceding section. By (the first part of) Proposition 7.5, it is easy to see that a pair (u, A) in \(W^{1,2}\) with
is a critical point for E (with respect to smooth perturbations supported in \(\Omega \)) if and only if the equations
are satisfied distributionally in \(\Omega \), where all geometric quantities and operators are defined with respect to the metric g.
Now, given a pair (u, A) in \(W^{1,2}\) satisfying (A.2)–(A.3) and
we can select a local Coulomb gauge adapted to \(\Omega \) as follows. Denote by \(\theta \in W^{2,2}(\Omega ,{\mathbb {R}})\) the unique solution of the Neumann problem
with zero mean \(\int _{\Omega }\theta =0\). Then the gauge-transformed pair
lies in \(W^{1,2}\) and continues to satisfy (A.2)–(A.3), with
but now with the additional constraints
For the remainder of the section, we will assume that the pair (u, A) is already in the Coulomb gauge on \(\Omega \), so that A satisfies (A.6). Note that (A.2)–(A.3) then become
We now establish the local regularity for critical points (u, A) in the Coulomb gauge, giving in particular local estimates for (u, A) in \(W^{2,q}\) norms.
Proposition A.1
Let (u, A) solve (A.2)–(A.3) in the Coulomb gauge (A.6) on \((\Omega ,g)\), with \(|u|\le 1\). If
and
then for every compactly supported subdomain \(\Omega '\subset \subset \Omega \) and \(q\in (1,\infty )\) there exists \(C_q(\Lambda ,\Omega ,\Omega ')<\infty \) such that
Proof
To begin, note that (A.8) and standard Bochner–Weitzenböck identities give the (weak) subequation
for \(|A|^2\). On the other hand, as in Sect. 3, we also obtain from (A.3) the relation
Recalling that \(|u|\le 1\) and using Young’s inequality, we can combine (A.12)–(A.13) to find an estimate of the form
for any \(0<\alpha <1\).
By standard estimates for one-forms A satisfying (A.6) (see, e.g., [20, Theorem 4.8]), we have the global \(L^2\) bound
hence |u|, |A| are both bounded in \(W^{1,2}\) in terms of \(\Lambda \) (and \(\Omega \)).
Note that (A.8) gives a local \(W^{2,2}\) bound on A, by standard elliptic regularity. This, together with Sobolev embedding and (A.7), gives
for all \(\Omega _0\subset \subset \Omega \) and some \(1<p<2\), depending only on n. We need the following observation, stated and proved separately for the sake of clarity.
Lemma A.2
Defining \(f\in W^{1,2}(\Omega )\) by
we have the subequation
and, for all \(\Omega _0\subset \subset \Omega \),
Proof
Since \(u\in L^\infty \cap W^{1,2}\cap W^{2,p}_{loc}\) and \(A\in W^{2,2}_{loc}\), a standard approximation argument shows that \(|u|^2,|A|^2\in W^{2,1}_{loc}\), so that (A.14) holds pointwise a.e.
Likewise, we have \(f\in W^{2,1}_{loc}\) and the chain rule applies, giving
pointwise. The first term equals \(f^{-1}\Delta \frac{1}{2}f^2\), so recalling (A.14) we obtain
Also, since \(f\in W^{1,2}\cap W^{2,p}_{loc}\), we have the pointwise inequalities
Choosing \(\alpha :=2-p\), inequality (A.16) follows. The second claim is an easy consequence of (A.15) and the fact that \(|u|\le 1\). \(\square \)
Returning to the proof of Proposition A.1, we can now apply Moser iteration to (A.16), obtaining in particular that
for any \(\Omega _1\subset \subset \Omega \).
Now, fixing some intermediate domain \(\Omega '\subset \subset \Omega _1\subset \subset \Omega \) between \(\Omega '\) and \(\Omega \), (A.7) together with the \(L^{\infty }(\Omega _1)\) estimate for A give pointwise bounds of the form
And since
in \(\Omega _1\), we obtain from the energy bound \(E(u,A)\le \Lambda \) and (A.18) the simple estimate
and consequently
for any \(\Omega '\subset \subset \Omega _2\subset \subset \Omega _1\). Returning to the pointwise bound (A.18), we can now employ a simple iteration argument—combining \(L^q\) regularity theory with the Sobolev embedding \(W^{2,r}\hookrightarrow W^{1,\frac{rn}{n-r}}\)—over successive domains between \(\Omega '\) and \(\Omega \), to arrive at the desired \(W^{2,q}\) estimates for u.
Returning finally to (A.8), it therefore follows from the preceding estimates that
for some intermediate domain \(\Omega '\subset \subset \Omega ''\subset \subset \Omega \). In particular, this gives us upper bounds for \(\Vert \Delta A\Vert _{L^q(\Omega '')}\) for every \(q\in (1,\infty )\), and \(L^q\) regularity theory therefore gives us the desired estimates for A in \(W^{2,q}(\Omega ')\). \(\square \)
Finally, we remark that higher regularity of u and A in the Coulomb gauge follows in a standard way—e.g., via Schauder theory—from the \(W^{2,q}\) estimates obtained in the preceding proposition.
Remark A.3
With local regularity established, note that it is easy to find a globally smooth couple \(({\widetilde{u}},{{\widetilde{\nabla }}})\) gauge equivalent to any critical pair \((u,\nabla )\) for \(E_{\epsilon }\) on \(L\rightarrow M\). Indeed, for any critical pair \((u,\nabla )\) with \(u\in W^{1,2}\cap L^{\infty }\) and \(\nabla =\nabla _0-iA\) (where \(\nabla _0\) is a smooth reference connection and \(A\in W^{1,2}\)), it follows from the local regularity results above that the gauge-invariant objects \(|u|^2\) and \(dA=F_{\nabla }-F_{\nabla _0}\) are smooth globally. Making a change of gauge \((u,\nabla )\rightarrow ({\widetilde{u}},{\widetilde{\nabla }}=\nabla _0-i{\widetilde{A}})\) such that
it follows from the smoothness of dA that the new connection \({\widetilde{\nabla }}=\nabla _0-i{\widetilde{A}}\) is smooth. And since \({\widetilde{u}}\) satisfies
where both \({\widetilde{\nabla }}\) and \(|u|^2\) are smooth, standard results for linear elliptic equations imply that \({\widetilde{u}}\in \Gamma (L)\) is a smooth section as well.
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Pigati, A., Stern, D. Minimal submanifolds from the abelian Higgs model. Invent. math. 223, 1027–1095 (2021). https://doi.org/10.1007/s00222-020-01000-6
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DOI: https://doi.org/10.1007/s00222-020-01000-6