Abstract
We study the limiting behavior as \(p\uparrow 2\) of the singular sets \(Sing(u_p)\) and p-energy measures \(\mu _p:=(2-p)|du_p|^pdvol\) for families of stationary p-harmonic maps \(u_p\in W^{1,p}(M,S^1)\) from a closed, oriented manifold M to the circle. When the measures \(\mu _p\) have uniformly bounded mass, we show that—up to subsequences—the singular sets \(Sing(u_p)\) converge in the Hausdorff sense to the support of a stationary, rectifiable varifold V of codimension 2, and the measures \(\mu _p\) converge weakly in \((C^0(M))^*\) to a limit of the form
where h is a harmonic one-form. For solutions on two-dimensional domains, we show moreover that the density of V takes values in \(2\pi {\mathbb {N}}\). Finally, we observe that nontrivial families \(u_p\) of such maps arise naturally on any closed Riemannian manifold of dimension \(n\ge 2\), via variational methods.
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Acknowledgements
The author thanks Fernando Codá Marques for his constant support and encouragement, and thanks the anonymous referee for their careful reading of the manuscript and valuable comments, which led to several improvements in the present version. During the completion of this project, the author was partially supported by NSF Grants DMS-1502424 and DMS-1509027.
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Appendix
Appendix
1.1 Proof of Proposition 2.2
In this short section, we demonstrate the independence from the parameter \(p\in [\frac{3}{2},2]\) of some standard estimates for p-harmonic functions (namely, the Lipschitz and \(W^{2,p}\) estimates discussed in Proposition 2.2). This is simply a matter of keeping track of p in the estimates of [14, 23], but we give some details in the interest of completeness.
Let \(B_2(x)\) be a geodesic ball in a manifold \(M^n\) satisfying the sectional curvature bound
and let \(\varphi \in W^{1,p}(B_2(x),{\mathbb {R}})\) be a p-harmonic function on \(B_2(x)\) for \(p\in [\frac{3}{2},2]\). Recall that, by the convexity of the p-energy functional, \(\varphi \) must be the unique minimizer for the p-energy with respect to its Dirichlet data.
For \(\epsilon >0\), we consider as in [23] the perturbed p-energy functionals
and let \(\varphi _{\epsilon }\in W^{1,p}(B_2(x))\) minimize \(F_{\epsilon }(\psi )\) with respect to the condition \(\psi -\varphi \in W_0^{1,p}(B_2(x))\). Setting
we then have that
and by standard results on quasilinear equations of this form (see, e.g., Chapter 4 of [22]), it follows that \(\varphi _{\epsilon }\) is a smooth, classical solution of (8.1). Moreover, since \(\varphi \) is the unique p-energy minimizer with respect to its Dirichlet data, we know that \(\varphi _{\epsilon }\rightarrow \varphi \) strongly in \(W^{1,p}(B_2)\) as \(\epsilon \rightarrow 0\). The task now (as in [14, 23]) is to establish estimates of the form given in (2.2) for the perturbed solutions \(\varphi _{\epsilon }\), and pass them to the limit \(\epsilon \rightarrow 0\).
As in [23], we observe now that, for \(\varphi _{\epsilon }\) solving (8.1), the energy density \(\gamma _{\epsilon }^p\) satisfies the divergence-form equation
where \(Hess(\varphi _{\epsilon })^2\) denotes the composition
and
In particular, it follows that
Now, since \(|\nabla \gamma _{\epsilon }|\le |Hess(\varphi _{\epsilon })|,\) when we integrate (8.4) against a test function \(\psi \in C_c^{\infty }(B_2(x))\) with \(\psi \equiv 1\) on \(B_1(x)\) and \(|\nabla \psi |\le 2\), we find that
and an application of Young’s inequality yields
In particular, since Hölder’s inequality gives
it follows that
and since \(p\in [\frac{3}{2},2]\), we can rewrite this as
To obtain \(L^{\infty }\) estimates for \(\gamma _{\epsilon }\), we can apply Moser iteration (see, e.g., [16], Chapter 8) to (8.4). Since the eigenvalues of
are bounded between \(p-1\) and 1, and we are working with \(p\in [\frac{3}{2},2]\), it is easy to see that the resulting estimate has the desired form
Finally, since \(\varphi _{\epsilon }\rightarrow \varphi \) strongly in \(W^{1,p}(B_2(x))\), we have that
and it follows from (8.7) and (8.6) that
and
Proposition 2.2 then follows by scaling.
1.2 Proof of Lemma 3.5
In this section, we prove Lemma 3.5, which we employed in the proof of Corollary 3.6. For convenience, we restate the lemma here:
Lemma 8.1
Let \(B_2(x)\) be a geodesic ball in a manifold \(M^n\) of sectional curvature \(|sec(M)|\le k\) and injectivity radius \(inj(M)\ge 3\). Let S be an \((n-2)\)-current in \(W^{-1,p}(B_2(x))\) satisfying, for some constant A,
for every ball \(B_r(y)\subset B_2(x)\). Suppose also that the r-tubular neighborhoods \({\mathcal {N}}_r(spt(S))\) about the support of S satisfy
Then there is a constant C(n, k, A) such that for every \(1<q<p\), we have
Let’s begin now by making some simple reductions. First, let’s assume that \(|sec(M)|\le 1\), since the result for a general sectional curvature bound \(|sec(M)|\le k\) then follows from a scaling and covering argument. We may then apply the Rauch comparison theorem to conclude that the given metric g on \(B_2(x)\) is uniformly equivalent to the flat one \(g_0\), with
for some constant C(n); and as a consequence, we see that it will suffice to establish the lemma in the flat case. Next, we note that every \((n-2)\)-current S in \(B^n_2(0)\subset {\mathbb {R}}^n\) is described by a finite collection of scalar distributions \(S_{ij}\), where
Thus, it is enough to show that Lemma 8.1 holds with a scalar distribution f in place of the \((n-2)\)-current S. Our first step in proving this is then the following observation:
Lemma 8.2
For \(p\in (1,2)\), let \(f\in W^{-1,p}(B_2^n(0))\) be a distribution on \(B_2^n(0)\) satisfying the estimate
for every ball \(B_r(x)\subset B_2^n\). Fixing a cutoff function \(\chi \in C_c^{\infty }(B_{5/3}(0))\) such that \(\chi \equiv 1\) on \(B_{4/3}(0)\), set
where G is the n-dimensional Euclidean Green’s function. We then have for \(x\in B_1(0)\setminus spt(f)\) a pointwise gradient estimate of the form
Proof
For \(x\in B_1\setminus spt(f)\), we observe that the pointwise derivatives \(w_i(x):=\partial _iw(x)\) are well-defined, and given by
where \(c_n\) is a dimensional constant.
To establish (8.14), first choose a function \(\zeta \in C_c^{\infty }([\frac{1}{2},2])\) satisfying
and for \(j\in {\mathbb {Z}}\), set
Defining
it’s easy to see that the functions \(\eta _j\) satisfy
and
Given \(x\in B_1^n\setminus spt(f)\), let \(m=\lceil -\log _2dist(x,spt(f))\rceil \), so that
Writing
and observing that
when \(y\in spt(\chi f)\subset B_4(x)\setminus B_{2^{-m}}(x),\) it follows that
Setting
we can then use (8.15)-(8.17) to see that
and
By (8.13), it therefore follows that
Summing from \(j=-m\) to \(j=2\), we obtain finally
giving the desired estimate (8.14). \(\square \)
Corollary 8.3
Let \(f\in W^{-1,p}(B_2^n(0))\) be as in Lemma 8.2, satisfying
for every ball \(B_r(x)\subset B_2(0)\). In addition, suppose that the tubular neighborhoods \({\mathcal {N}}_r(spt(f))\) about the support of f satisfy the volume bound
Then there is a constant \(C(n,A)<\infty \) depending only on n and A such that for every \(q \in (1,p)\), we have the estimate
Proof
By Lemma 8.2, there exists a function \(w\in W^{1,p}(B_2^n(0))\) satisfying
and
for \(x\in B_1(0)\setminus spt(f)\). For any \(\varphi \in C_c^{\infty }(B_1(0))\) and \(q\in (1,p)\), we then have
while, by (8.21) and (8.19), we see that
Thus, we indeed have
the desired \(W^{-1,q}\) estimate. \(\square \)
As remarked previously, Lemma 8.1 now follows by applying Corollary 8.3 to the scalar component distributions of the \((n-2)\)-current S.