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p-Harmonic maps to \(S^1\) and stationary varifolds of codimension two

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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

$$\begin{aligned} \mu =\Vert V\Vert +|h|^2dvol, \end{aligned}$$

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

$$\begin{aligned} |sec(M)|\le k, \end{aligned}$$

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

$$\begin{aligned} F_{\epsilon }(\psi )=\int (\epsilon +|d\psi |^2)^{p/2}, \end{aligned}$$

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

$$\begin{aligned} \gamma _{\epsilon }:=(\epsilon +|d\varphi _{\epsilon }|^2)^{1/2}, \end{aligned}$$

we then have that

$$\begin{aligned} div(\gamma _{\epsilon }^{p-2}d\varphi _{\epsilon })=0, \end{aligned}$$
(8.1)

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

$$\begin{aligned} div(A_{\epsilon }\nabla (\gamma _{\epsilon }^p))=p\gamma _{\epsilon }^{p-2}[\langle A_{\epsilon },Hess(\varphi _{\epsilon })^2\rangle +Ric(d\varphi _{\epsilon },d\varphi _{\epsilon })], \end{aligned}$$
(8.2)

where \(Hess(\varphi _{\epsilon })^2\) denotes the composition

$$\begin{aligned} Hess(\varphi _{\epsilon })^2(X,Y)=tr(Hess(\varphi _{\epsilon })(X,\cdot )Hess(\varphi _{\epsilon })(Y,\cdot )), \end{aligned}$$

and

$$\begin{aligned} A_{\epsilon }:=I+(p-2)\gamma _{\epsilon }^{-2}d\varphi _{\epsilon }\otimes d\varphi _{\epsilon }. \end{aligned}$$
(8.3)

In particular, it follows that

$$\begin{aligned} div(A_{\epsilon }\nabla (\gamma _{\epsilon }^p))\ge p(p-1)\gamma _{\epsilon }^{p-2}|Hess(\varphi _{\epsilon })|^2-C(n,k)\gamma _{\epsilon }^p. \end{aligned}$$
(8.4)

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

$$\begin{aligned} \int \psi ^2 p(p-1)\gamma _{\epsilon }^{p-2}|Hess(\varphi _{\epsilon })|^2\le & {} \int 2p\psi |d\psi |\gamma _{\epsilon }^{p-1}|\nabla \gamma _{\epsilon }|+C(k,n)\gamma _{\epsilon }^p\\\le & {} \int _{B_2} 4p\gamma _{\epsilon }^{p/2}(\psi \gamma _{\epsilon }^{\frac{p-2}{2}}|Hess(\varphi _{\epsilon })|)+C(k,n)\gamma _{\epsilon }^p, \end{aligned}$$

and an application of Young’s inequality yields

$$\begin{aligned} p(p-1)\int \psi ^2 \gamma _{\epsilon }^{p-2}|Hess(\varphi _{\epsilon })|^2\le \frac{C(k,n)}{(p-1)}\int _{B_2}\gamma _{\epsilon }^p. \end{aligned}$$
(8.5)

In particular, since Hölder’s inequality gives

$$\begin{aligned} \int _{B_1} |Hess(\varphi _{\epsilon })|^p\le \left( \int _{B_1}\gamma _{\epsilon }^{p-2}|Hess(\varphi _{\epsilon })|^2\right) ^{p/2}\left( \int \gamma _{\epsilon }^p\right) ^{\frac{2-p}{2}}, \end{aligned}$$

it follows that

$$\begin{aligned} \Vert d\varphi _{\epsilon }\Vert ^p_{W^{1,p}(B_1)}\le \frac{C(k,n)}{(p-1)^2}\int _{B_2}\gamma _{\epsilon }^p, \end{aligned}$$

and since \(p\in [\frac{3}{2},2]\), we can rewrite this as

$$\begin{aligned} \Vert d\varphi _{\epsilon }\Vert ^p_{W^{1,p}(B_1)}\le C(k,n)\int _{B_2}\gamma _{\epsilon }^p. \end{aligned}$$
(8.6)

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

$$\begin{aligned} A_{\epsilon }=I+(p-2)\gamma _{\epsilon }^{-2}d\varphi _{\epsilon }\otimes d\varphi _{\epsilon } \end{aligned}$$

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

$$\begin{aligned} \Vert d\varphi _{\epsilon }\Vert _{L^{\infty }(B_1)}^p\le \Vert \gamma _{\epsilon }\Vert _{L^{\infty }(B_1)}^p\le C(k,n)\int _{B_2}\gamma _{\epsilon }^p. \end{aligned}$$
(8.7)

Finally, since \(\varphi _{\epsilon }\rightarrow \varphi \) strongly in \(W^{1,p}(B_2(x))\), we have that

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}\int _{B_2}\gamma _{\epsilon }^p=\int _{B_2} |d\varphi |^p, \end{aligned}$$

and it follows from (8.7) and (8.6) that

$$\begin{aligned} \Vert d\varphi \Vert ^p_{L^{\infty }(B_1)}\le \liminf _{\epsilon \rightarrow 0}\Vert d\varphi _{\epsilon }\Vert _{L^{\infty }(B_1)}^p\le C(k,n)\int _{B_2}|d\varphi |^p, \end{aligned}$$
(8.8)

and

$$\begin{aligned} \Vert d\varphi \Vert _{W^{1,p}(B_1)}^p\le \liminf _{\epsilon \rightarrow 0}\Vert d\varphi _{\epsilon }\Vert ^p_{W^{1,p}(B_1)}\le C(k,n)\int _{B_2}|d\varphi |^p. \end{aligned}$$
(8.9)

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,

$$\begin{aligned} \langle S,\zeta \rangle \le A r^{n-p}\Vert \zeta \Vert ^{p-1}_{L^{\infty }}\Vert d\zeta \Vert ^{2-p}_{L^{\infty }} {}\forall \zeta \in \Omega _c^{n-2}(B_r(y)) \end{aligned}$$
(8.10)

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

$$\begin{aligned} Vol(B_2(x)\cap {\mathcal {N}}_r(spt(S)))\le Ar^p. \end{aligned}$$
(8.11)

Then there is a constant C(nkA) such that for every \(1<q<p\), we have

$$\begin{aligned} \Vert S\Vert _{W^{-1,q}(B_1(x))}\le C(n,k,A)(p-q)^{-1/q}. \end{aligned}$$
(8.12)

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

$$\begin{aligned} C(n)^{-1}g_0\le g\le C(n)g_0 \end{aligned}$$

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

$$\begin{aligned} \langle S_{ij},\varphi \rangle :=\langle S, *\varphi dx^i\wedge dx^j\rangle . \end{aligned}$$

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

$$\begin{aligned} \langle f,\varphi \rangle \le Ar^{n-p}\Vert \varphi \Vert _{L^{\infty }}^{p-1}\Vert d\varphi \Vert _{L^{\infty }}^{2-p} { }\forall \varphi \in C_c^{\infty }(B_r(x)) \end{aligned}$$
(8.13)

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

$$\begin{aligned} w(x):=\langle (\chi f)(y),G(x-y)\rangle , \end{aligned}$$

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

$$\begin{aligned} |dw(x)|\le C_n A\cdot dist(x,spt(f))^{-1}. \end{aligned}$$
(8.14)

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

$$\begin{aligned} w_i(x):=\partial _iw(x)=c_n\langle (\chi f)(y), |x-y|^{-n}(x-y)_i\rangle , \end{aligned}$$

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

$$\begin{aligned} \zeta \equiv 1\text { on }[\frac{3}{4},\frac{3}{2}]\text { and }|\zeta '|\le 10, \end{aligned}$$

and for \(j\in {\mathbb {Z}}\), set

$$\begin{aligned} \zeta _j(t):=\zeta (2^{-j}t). \end{aligned}$$

Defining

$$\begin{aligned} \eta _j(t):=\frac{\zeta _j(t)}{\Sigma _{k\in {\mathbb {Z}}}\zeta _k(t)}, \end{aligned}$$

it’s easy to see that the functions \(\eta _j\) satisfy

$$\begin{aligned}&spt(\eta _j)\subset (2^{j-1},2^{j+1}), \end{aligned}$$
(8.15)
$$\begin{aligned}&\Sigma _{j\in {\mathbb {Z}}}\eta _j(t)=1, \end{aligned}$$
(8.16)

and

$$\begin{aligned} |\eta _j'|\le 30\cdot 2^{-j}. \end{aligned}$$
(8.17)

Given \(x\in B_1^n\setminus spt(f)\), let \(m=\lceil -\log _2dist(x,spt(f))\rceil \), so that

$$\begin{aligned} 2^{1-m}\ge dist(x,spt(f))\ge 2^{-m}. \end{aligned}$$

Writing

$$\begin{aligned} w_i(x)= & {} c_n\langle (\chi f)(y),|x-y|^{-n}(x-y)_i\rangle \\= & {} c_n\langle (\chi f)(y),\Sigma _{j\in {\mathbb {Z}}}\eta _j(|x-y|)|x-y|^{-n}(x-y)_i\rangle , \end{aligned}$$

and observing that

$$\begin{aligned} 1-\Sigma _{j=-m}^2\eta _j(|x-y|)=0 \end{aligned}$$

when \(y\in spt(\chi f)\subset B_4(x)\setminus B_{2^{-m}}(x),\) it follows that

$$\begin{aligned} w_i(x)= & {} c_n\langle (\chi f)(y),\Sigma _{j=-m}^2\eta _j(|x-y|)|x-y|^{-n}(x-y)_i\rangle \\= & {} c_n\Sigma _{j=-m}^2\langle (\chi f)(y),\eta _j(|x-y|)|x-y|^{-n}(x-y)_i\rangle . \end{aligned}$$

Setting

$$\begin{aligned} \varphi _j(y):=\chi (y)\eta _j(|x-y|)|x-y|^{-n}(x-y)_i, \end{aligned}$$

we can then use (8.15)-(8.17) to see that

$$\begin{aligned}&spt(\varphi _j)\subset B_{2^{j+1}}(x),\\&\Vert \varphi _j\Vert _{L^{\infty }}\le 2^{(j-1)(1-n)}, \end{aligned}$$

and

$$\begin{aligned} \Vert d\varphi _j\Vert _{L^{\infty }}\le C_n2^{-n(j-1)}. \end{aligned}$$

By (8.13), it therefore follows that

$$\begin{aligned} |\langle f,\varphi _j\rangle |\le & {} A(2^{j+1})^{n-p}\Vert \varphi _j\Vert _{L^{\infty }}^{p-1}\Vert d\varphi _j\Vert _{L^{\infty }}^{2-p}\\\le & {} C_nA(2^{j+1})^{n-p}\cdot 2^{(j-1)(1-n)(p-1)}\cdot 2^{-n(2-p)(j-1)}\\\le & {} C_n' A 2^{-j}. \end{aligned}$$

Summing from \(j=-m\) to \(j=2\), we obtain finally

$$\begin{aligned} |w_i(x)|= & {} |c_n\Sigma _{j=-m}^2\langle f, \varphi _j(y)\rangle |\\\le & {} C_nA\Sigma _{j=-m}^2 2^{-j}\\\le & {} C_n A 2^m\\\le & {} 2C_n A \cdot dist(x,spt(f))^{-1}, \end{aligned}$$

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

$$\begin{aligned} \langle f,\varphi \rangle \le Ar^{n-p}\Vert \varphi \Vert _{L^{\infty }}^{p-1}\Vert d\varphi \Vert _{L^{\infty }}^{2-p} \text { }\forall \varphi \in C_c^{\infty }(B_r(x)) \end{aligned}$$
(8.18)

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

$$\begin{aligned} Vol({\mathcal {N}}_r(spt(f)))\le Ar^p. \end{aligned}$$
(8.19)

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

$$\begin{aligned} \Vert f\Vert _{W^{-1,q}(B_1(0))}\le C(n,A)(p-q)^{-1/q}. \end{aligned}$$
(8.20)

Proof

By Lemma 8.2, there exists a function \(w\in W^{1,p}(B_2^n(0))\) satisfying

$$\begin{aligned} \Delta w=f\text { on }B_{4/3}(0) \end{aligned}$$

and

$$\begin{aligned} |dw(x)|\le \frac{C_nA}{dist(x,spt(f))} \end{aligned}$$
(8.21)

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

$$\begin{aligned} \langle f,\varphi \rangle= & {} \langle \Delta w, \varphi \rangle \\= & {} -\,\int \langle dw,d\varphi \rangle \\\le & {} \Vert dw\Vert _{L^q(B_1(0))}\Vert d\varphi \Vert _{L^{q'}}, \end{aligned}$$

while, by (8.21) and (8.19), we see that

$$\begin{aligned} \int _{B_1(0)}|dw|^q\le & {} C_nA^q\int _{B_1(0)}dist(x,spt(f))^{-q}\\ \text { (since }dist(x,spt(f))\le 3\text { on }B_1(0))\le & {} 4C_nA^q\int _{B_1(0)}[dist(x,spt(f))^{-q}-4^{-q}]\\ \text { (by the fundamental theorem of calculus) }= & {} CA^q\int _{B_1(0)}\int _{dist(x,spt(f))}^4qr^{-q-1}dr\\ \text { (by Fubini) }= & {} CA^q\int _0^4q r^{-q-1}Vol({\mathcal {N}}_r(spt(f)))dr\\\le & {} C A^{q+1}\int _0^4 r^{p-q-1}dr\\\le & {} \frac{C A^{q+1}}{p-q}. \end{aligned}$$

Thus, we indeed have

$$\begin{aligned} \langle f,\varphi \rangle \le C(n,A)(p-q)^{-1/q}\Vert d\varphi \Vert _{L^{q'}}, \end{aligned}$$

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.

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Stern, D. p-Harmonic maps to \(S^1\) and stationary varifolds of codimension two. Calc. Var. 59, 187 (2020). https://doi.org/10.1007/s00526-020-01859-6

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