The Viscosity Method for Min–Max Free Boundary Minimal Surfaces

Abstract

We adapt the viscosity method introduced by Rivière (Publ Math Inst Hautes Études Sci 126:177–246, 2017. https://doi.org/10.1007/s10240-017-0094-z) to the free boundary case. Namely, given a compact oriented surface $\Sigma$, possibly with boundary, a closed ambient Riemannian manifold $({\mathcal {M}}^m,g)$ and a closed embedded submanifold ${\mathcal {N}}^n\subset {\mathcal {M}}$, we study the asymptotic behavior of (almost) critical maps $\Phi$ for the functional

$$\begin{aligned} E_\sigma (\Phi ):={\text {area}}(\Phi )+\sigma {\text {length}}(\Phi |_{\partial \Sigma })+\sigma ^4\int _\Sigma |\mathrm {I\!I}^\Phi |^4\,{\text {vol}}_\Phi \end{aligned}$$

on immersions $\Phi :\Sigma \rightarrow {\mathcal {M}}$ with the constraint $\Phi (\partial \Sigma )\subseteq {\mathcal {N}}$, as $\sigma \rightarrow 0$, assuming an upper bound for the area and a suitable entropy condition. As a consequence, given any collection ${\mathcal {F}}$ of compact subsets of the space of smooth immersions $(\Sigma ,\partial \Sigma )\rightarrow ({\mathcal {M}},{\mathcal {N}})$, assuming ${\mathcal {F}}$ to be stable under isotopies of this space, we show that the min–max value

$$\begin{aligned} \inf _{A\in {\mathcal {F}}}\max _{\Phi \in A}{\text {area}}(\Phi ) \end{aligned}$$

is the sum of the areas of finitely many branched minimal immersions $\Phi _{(i)}:\Sigma _{(i)}\rightarrow {\mathcal {M}}$ with $\Phi _{(i)}(\partial \Sigma _{(i)})\subseteq {\mathcal {N}}$ and $\partial _\nu \Phi _{(i)}\perp T{\mathcal {N}}$ along $\partial \Sigma _{(i)}$, whose (connected) domains $\Sigma _{(i)}$ can be different from $\Sigma$ but cannot have a more complicated topology. Contrary to other min–max frameworks, the present one applies in an arbitrary codimension. We adopt a point of view which exploits extensively the diffeomorphism invariance of $E_\sigma$ and, along the way, we simplify and streamline several arguments from the initial work (Rivière 2017). Some parts generalize to closed higher-dimensional domains, for which we get an integral stationary varifold in the limit.

Appendix

Proposition A.1

A continuous, $W^{1,2}$ map $u:B_1^2\rightarrow {\mathbb {R}}^m$ solving a linear system of the form

$$\begin{aligned} -\partial _i(g_{jk}\partial _i u^j)+b_{kpq}\partial _i u^p\partial _i u^q=0, \end{aligned}$$

(A.1)

with $g\geqq \lambda >0$ symmetric and continuous and b bounded, is $W^{1,r}_{loc}$ for all $r<\infty$.

The same holds for u defined on the half-ball $U':=B_1^2\cap \{\Im (z)\geqq 0\}$, if in addition we have

$$\begin{aligned} \partial _\nu u^k=0\text { for }k\leqq n,\quad u^k=0\text { for }k>n, \end{aligned}$$

as well as $g_{ij}=0$ for $i\leqq n$, $j>n$, on the boundary $\partial U'$, for some $0\leqq n\leqq m$.

Remark A.2

The condition $\partial _\nu u^k=0$ could be written more faithfully as $g_{jk}\partial _\nu \Phi ^j=0$ and is of course meant in a weak sense, coupled with the equation: namely, we require $\int _{U'}(g_{jk}\partial _i f\partial _i u^j+b_{kpq}f\partial _i u^p \partial _i u^q)=0$ for all $f\in C^\infty _c(U')$ and $k\leqq n$, allowing f to be nonzero on $\partial U'$.

Proof

Assume u is a solution on the unit ball. Then, for any ball $B_{2r}^2(x)\subseteq B_1^2$, we can integrate the equation against $\eta ^2(u-(u)_{B_{2r}^2(x)})$, where $\eta \in C^\infty _c(B_{2r}^2(x))$ is a cut-off function satisfying $\eta =1$ on $B_r^2(x)$ and $|\mathrm{d}\eta |\leqq \frac{2}{r}$. Recall that the notation $(u)_S$ indicates the average of u on a set S. This gives

$$\begin{aligned} \lambda \int \eta ^2|\mathrm{d}u|^2 \leqq C\int \eta |\mathrm{d}u|\,|\mathrm{d}\eta |\,|u-(u)_{B_{2r}^2(x)}| +C\int \eta ^2|\mathrm{d}u|^2{\text {osc}}(u,B_{2r}^2(x)) \end{aligned}$$

and, applying Young’s inequality, it follows that

$$\begin{aligned} \int _{B_r^2(x)}|\mathrm{d}u|^2\leqq Cr^{-2}\int _{B_{2r}^2(x)}|u-(u)_{B_{2r}^2(x)}|^2 \leqq Cr^{-2}\left( \int _{B_{2r}^2(x)}|\mathrm{d}u|\right) ^2 \end{aligned}$$

whenever ${\text {osc}}(u,B_{2r}^2(x))$ is small enough. The classical Gehring’s lemma (see, for example, [17, Theorem V.1.2]) then implies that $\mathrm{d}u\in L^r(B)$ for some $r>2$ and any fixed ball $B\subset \subset B_1^2$ (with r depending on B). Then the nonlinear term $b_{kpq}\partial _i u^p\partial _i u^q$ is $L^{r/2}(B)$ and standard elliptic regularity theory gives $\mathrm{d}u\in L^s_{loc}(B)$, with $\frac{1}{s}=\frac{2}{r}-\frac{1}{2}$, so that $s>r$; iterating, we get $\mathrm{d}u\in L^t_{loc}$ for any $t<\infty$.

If we are in the half-ball case, then we can reduce to the previous case by reflection. We extend g and u to ${\tilde{g}}$ and ${\tilde{u}}$ on the ball $B_1^2$, by means of the formula

$$\begin{aligned} g(s,-t):=Ug(s,t)U,\quad \begin{pmatrix}{\tilde{u}}^1 \\ \vdots \\ {\tilde{u}}^m\end{pmatrix}(s,-t):=U\begin{pmatrix}u^1 \\ \vdots \\ u^m\end{pmatrix}(s,t) \end{aligned}$$

for $(s,-t)$ in the lower half-ball, with $U:=\begin{pmatrix}I_n &{}\quad \\ &{}\quad -I_{m-n}\end{pmatrix}$. Note that, by our hypotheses on g, ${\tilde{g}}$ is still continuous. Also, it is straightforward to check that ${\tilde{u}}$ solves

$$\begin{aligned} -\partial _i({\tilde{g}}_{jk}\partial _i {\tilde{u}}^j)+{\tilde{b}}_{kpq}\partial _i {\tilde{u}}^p\partial _i {\tilde{u}}^q=0, \end{aligned}$$

with ${\tilde{b}}_{kpq}$ extending $b_{kpq}$ according to the following rule: if $k\leqq n$ then ${\tilde{b}}_{kpq}(s,-t):=b_{kpq}(s,t)$ if p and q belong to the same set in the partition $\{\{1,\dots ,n\},\{n+1,\dots ,m\}\}$, and ${\tilde{b}}_{kpq}(s,-t):=-b_{kpq}(s,t)$ otherwise; if $k>n$ then the opposite holds. Then from the case of the full ball we deduce $\mathrm{d}{\tilde{u}}\in L^t_{loc}$ for any $t<\infty$. $\square$

Remark A.3

If the coefficients in (A.1) are smooth functions of u, then u is smooth. To check this, note that in the full ball case u is $C^{0,\alpha }_{loc}$ for any $\alpha <1$. The same is then true for the coefficients $g_{jk}(u)$. Since the nonlinearity $b_{kpq}\partial _i u^p\partial _i u^q$ belongs to $L^r_{loc}$ for all $r<\infty$, classical Schauder theory then gives $\mathrm{d}u\in C^{0,\alpha }_{loc}$ for all $\alpha <1$ and bootstrapping we reach $u\in C^\infty$.

In the half-ball case, we can still argue in the same way that $\mathrm{d}{\tilde{u}}\in C^{0,\alpha }_{loc}$ for all $\alpha <1$. So ${\tilde{g}}$ is locally Lipschitz and we deduce ${\tilde{u}}\in W^{2,r}_{loc}$ for all $r<\infty$. Differentiating the original equation in the first variable preserves the boundary conditions and leads to an equation of the form

$$\begin{aligned} \partial _i(g_{jk}\partial _i(\partial _1 u^j))+f_k=0 \end{aligned}$$

with $f_k\in L^r_{loc}$ for all $r<\infty$, and the same reflection trick (applied to $w:=\partial _1 u$) gives $\partial _1 u\in W^{2,r}_{loc}$ for all $r<\infty$. Iterating we get the same for all derivatives $\partial _1^k u$. Now the equation allows to deduce inductively that $u\in W^{k,r}_{loc}$ for all k, since $g_{jk}(u)\Delta u^j=-\partial _i(g_{jk}(u))\partial _i u^j+b_{kpq}(u)\partial _i u^p\partial _i u^q$; this expresses $\partial _{22}u$ in terms of $\partial _{11}u$ and lower order derivatives and hence, for any multi-index $\alpha =(\alpha _1,\alpha _2)$ with $\alpha _2\geqq 2$, we deduce that $\partial ^\alpha u=\partial _1^{\alpha _1}\partial _2^{\alpha _2}u\in L^r_{loc}$ for all $r<\infty$ from the same property enjoyed by $\partial _1^{\alpha _1+2}\partial _2^{\alpha _2-2}u$ and lower order derivatives of u.

The recent statements deal with general varifolds. It is clear that we can assume the smallness constant $c_V$ appearing in all of them to be always the same.

Lemma A.4

There exists $c_V({\mathcal {M}},{\mathcal {N}})>0$ with the following property. Given $p\in {\mathcal {N}}$ and $0<s<c_V$, for any 2-varifold ${\mathbf {v}}$ on ${\mathcal {M}}$ which is free boundary stationary outside ${\bar{B}}_s(p)$ and has density $\theta \geqq {\bar{\theta }}$ on ${\text {spt}}(|{\mathbf {v}}|){\setminus }{\bar{B}}_s(p)$, either ${\text {spt}}(|{\mathbf {v}}|)\subseteq B_{2s}(p)$ or $|{\mathbf {v}}|({\mathcal {M}}{\setminus }{\bar{B}}_s(p))\geqq c_V{\bar{\theta }}$.

Proof

Pick $\gamma >0$ small, to be fixed along the proof; we will choose $c_V\leqq \gamma$, so that the varifold is free boundary stationary outside ${\bar{B}}_\gamma (p)$ Possibly multiplying ${\mathbf {v}}$ by ${\bar{\theta }}^{-1}$, we can assume ${\bar{\theta }}=1$. Note that if $q\in {\text {spt}}(|{\mathbf {v}}|){\setminus }B_{2\gamma }(p)$, then, by (4.8), we have

$$\begin{aligned} |{\mathbf {v}}|(B_\gamma (q)) \geqq c({\mathcal {M}},{\mathcal {N}})\gamma ^2\theta (|{\mathbf {v}}|,q) \geqq c({\mathcal {M}},{\mathcal {N}})\gamma ^2. \end{aligned}$$

(A.2)

Otherwise, $|{\mathbf {v}}|$ is supported in $B_{2\gamma }(p)$. Assume we are in this second case and pick a set of coordinates $(x_1,\dots ,x_m):B_{5\gamma }(p)\rightarrow {\mathbb {R}}^m$ centered at p, with ${\mathcal {N}}$ corresponding to $\{x_{n+1}=\dots =x_m=0\}$. We can impose that $\Vert g_{ij}-\delta _{ij}\Vert _{C^1}\leqq \gamma$ (in coordinates), for $\gamma$ small, independently of $p\in {\mathcal {N}}$.

On this ball, we define the vector field X to be $X(x):=\chi (|x|)x_i\frac{\partial }{\partial x_i}$, where $\chi :[0,\infty )\rightarrow [0,1]$ is smooth and such that $\chi '\geqq 0$ on $[0,3\gamma ]$, $\chi =1$ on $[\frac{5}{3}s,3\gamma ]$, $\chi =0$ on $[0,\frac{4}{3}s]\cup [4\gamma ,\infty )$. Assuming $\{|x|\leqq 4\gamma \}\subset \subset B_{5\gamma }(p)$, we can smoothly extend X to all of ${\mathcal {M}}$, with $X=0$ outside the ball. For $\gamma$ small enough (independently of p and $s<\gamma$), the $C^1$ closeness of $g_{ij}$ to $\delta _{ij}$ guarantees

$$\begin{aligned} {\text {div}}_\Pi X\geqq 0 \end{aligned}$$

for all $(p,\Pi )\in {\text {Gr}}_2({\mathcal {M}})$ in the support of ${\mathbf {v}}$, since we can assume ${\text {spt}}(|{\mathbf {v}}|)\subseteq \{|x|<3\gamma \}$: indeed, here the contribution of $\chi '$ is nonnegative, while the one of the position vector $x_i\frac{\partial }{\partial x_i}$ is close to 2 (multiplied by $\chi (|x|)$). Also, the inequality is strict if $|x(p)|\geqq \frac{5}{3}s$. Moreover, X is tangent to ${\mathcal {N}}$. We can also assume that ${\bar{B}}_s(p)\subset \subset \{|x|\leqq \frac{4}{3}s\}$; hence, we can test the stationarity of ${\mathbf {v}}$ against X and reach the contradiction

$$\begin{aligned} 0=\int _{(p,\Pi )\in {\text {Gr}}_2({\mathcal {M}})}{\text {div}}_\Pi X\,\mathrm{d}{\mathbf {v}}(p,\Pi )>0, \end{aligned}$$

unless ${\text {spt}}(|{\mathbf {v}}|)$ is contained in $\{|x|\leqq \frac{5}{3}s\}$. Since the latter can be assumed to be included in $B_{2s}(p)$, the statement follows from (A.2). $\square$

Remark A.5

The same statement holds if ${\mathbf {v}}$ is stationary outside ${\bar{B}}_s(p)$, without the assumption $p\in {\mathcal {N}}$. The proof is analogous (but simpler, in that we do not need coordinates adapted to ${\mathcal {N}}$).

Lemma A.6

There exist $c_V>0$ and $\delta :(0,\infty )^2\rightarrow (0,\infty )$, with $\lim _{s\rightarrow 0}\delta (s,t)=0$ for every t, satisfying the following property. Given two points $p_1,p_2\in {\mathcal {M}}$ and a radius $s>0$, let $B:={\bar{B}}_s(p_1)\cup {\bar{B}}_s(p_2)$; if a 2-varifold ${\mathbf {v}}$ on ${\mathcal {M}}$ is free boundary stationary outside B, has density $\theta \geqq {\bar{\theta }}$ on ${\text {spt}}(|{\mathbf {v}}|){\setminus }B$ and satisfies the bound

$$\begin{aligned} |{\mathbf {v}}|(B_r(q))\leqq c'r^2\quad \text {for all }q\in {\mathcal {M}},\ r>0, \end{aligned}$$

then either $|{\mathbf {v}}|({\mathcal {M}})\leqq {\bar{\theta }}\delta (s,c'/{\bar{\theta }})$ or $|{\mathbf {v}}|({\mathcal {M}})\geqq c_V{\bar{\theta }}$. The constant $c_V$ and the function $\delta$ depend only on ${\mathcal {M}}$ and ${\mathcal {N}}$.

Proof

We can assume ${\bar{\theta }}=1$. From (4.8) it follows that any nontrivial free boundary stationary varifold ${\mathbf {v}}'$ with density at least 1 on ${\text {spt}}(|{\mathbf {v}}'|)$ has $|{\mathbf {v}}'|({\mathcal {M}})\geqq \lambda ({\mathcal {M}},{\mathcal {N}})$. Let $\delta (s,c')$ be the supremum of all possible masses $|{\mathbf {v}}|({\mathcal {M}})$ which are smaller than $c_V$, for ${\mathbf {v}}$ as in the statement, with $c_V$ to be specified below. Take a sequence $s_k\rightarrow 0$ of positive numbers and a sequence ${\mathbf {v}}_k$ satisfying the assumptions with $s=s_k$, as well as $\delta (s_k,c')-2^{-k}<|{\mathbf {v}}_k|({\mathcal {M}})<c_V$.

Up to subsequences we get a limit varifold ${\mathbf {v}}_\infty$ which is free boundary stationary on the complement of two points ${\bar{p}}_1$ and ${\bar{p}}_2$. We still have $|{\mathbf {v}}_\infty |(B_r(q))\leqq c'r^2$ for all centers q and all radii r. This upper bound implies easily that actually ${\mathbf {v}}_\infty$ is free boundary stationary on the full manifold see the proof of Theorem 5.12 for the details; also, by (4.8), it has a lower bound $c\leqq 1$ for its density on ${\text {spt}}(|{\mathbf {v}}_\infty |)$. Hence, $|{\mathbf {v}}_\infty |({\mathcal {M}})\geqq c\lambda$ unless ${\mathbf {v}}_\infty =0$.

Since $|{\mathbf {v}}_\infty |({\mathcal {M}})=\lim _{k\rightarrow \infty }|{\mathbf {v}}_k|({\mathcal {M}})\leqq c_V$, choosing any $c_V<c\lambda$ forces ${\mathbf {v}}_\infty =0$, so that $\delta (s_k,c')\rightarrow 0$. This shows that $\delta (s,c')\rightarrow 0$ as $s\rightarrow 0$. $\square$