Multiplicity one for min–max theory in compact manifolds with boundary and its applications

Abstract

We prove the multiplicity one theorem for min–max free boundary minimal hypersurfaces in compact manifolds with boundary of dimension between 3 and 7 for generic metrics. To approach this, we develop existence and regularity theory for free boundary hypersurface with prescribed mean curvature, which includes the regularity theory for minimizers, compactness theory, and a generic min–max theory with Morse index bounds. As applications, we construct new free boundary minimal hypersurfaces in the unit balls in Euclidean spaces and self-shrinkers of the mean curvature flows with arbitrarily large entropy.

Appendices

Appendix A: Removing singularity for weakly stable free boundary h-hypersurfaces

Theorem A.1

(cf. [3, Theorem 27]) Let $(M^{n+1},\partial M,g)$ be a compact Riemannian manifold with boundary of dimension $3\le (n + 1)\le 7$. Given $h\in C^\infty (M)$ and $\Sigma \subset B_\epsilon (p){\setminus }\{p\}$ an almost embedded free boundary h-hypersurface with $\partial \Sigma \cap \textrm{Int}M\cap B_\epsilon (p){\setminus }\{p\} = \emptyset$, assume that $\Sigma$ is weakly stable in $B_\epsilon (p){\setminus }\{p\}$ as in Remark 2.7. If $[\Sigma ]$ represents a varifold of bounded first variation in $B_\epsilon (p)$, then $\Sigma$ extends smoothly across p as an almost embedded hypersurface in $B_\epsilon (p)$.

Proof

Given any sequence of positive $\lambda _i \rightarrow 0$, consider the blowups $\{\varvec{\mu }_{p,\lambda _i} (\Sigma ) \subset \varvec{\mu }_{p,\lambda _i} (M)\}$, where $\varvec{\mu }_{p,\lambda _i} (x) = \frac{x-p}{\lambda _i}$. Since $\Sigma$ has bounded first variation, $\varvec{\mu }_{p,\lambda _i} (\Sigma )$ converges (up to a subsequence) to a stationary integral rectifiable cone C in a half Euclidean space $\mathbb {R}^{n+1}_+ = T_p M$ with free boundary on $\partial \mathbb {R}^{n+1}_+=T_p(\partial M)$. By weak stability and Theorem 2.9, the convergence is locally smooth and graphical away from the origin, so C is an integer multiple of some embedded minimal hypercone with free boundary. Hence the reflection of C across $\partial \mathbb {R}^{n+1}_+$ is a stable minimal cone in $\mathbb {R}^{n+1}{\setminus } \{0\}$, and hence a plane with integer multiplicities. Therefore, we conclude that $C=m\cdot T_p(\partial M)$ or $2\,m\cdot P$ for some hyperplane $P\subset \mathbb {R}^{n+1}_+$ with $P\perp \partial \mathbb {R}^{n+1}_+$. Here $m=\Theta ^n(\Sigma ,p)$. Note that P may depend on the choice of $\{\lambda _i\}$.

If $C=m\cdot T_p(\partial M)$, then the argument in [60, Theorem B.1] implies the removability of p.

If $C=2m\cdot P$ for some half-hyperplane P, then by the locally smooth and graphical convergence, there exists $\sigma _0> 0$ small enough, such that for any $0 < \sigma \le \sigma _0$, $\Sigma$ has an m-sheeted, ordered, graphical decomposition in the annulus $A_{\sigma /2,\sigma }(p) =M\cap B_\sigma (p){\setminus } B_{\sigma /2} (p)$:

$$\begin{aligned}\Sigma \cap A_{\sigma /2,\sigma } (p) = \bigcup _{i=1}^m \Sigma _i(\sigma ).\end{aligned}$$

Here each $\Sigma _i (\sigma )$ is a graph over $A_{\sigma /2,\sigma } (p)\cap P$. We can continue each $\Sigma _i (\sigma )$ all the way to $B_{\sigma _0}(p){\setminus } \{p\}$, and denote the continuation by $\Sigma _i$. Then each $\Sigma _i$ is a free boundary h-hypersurface in $M\cap B_{\Sigma _0}(p){\setminus } \{p\}$ and $\Theta ^n(\Sigma _i,p)=1/2$. By the Allard type regularity theorem for rectifiable varifolds with free boundary and bounded mean curvature [19, Theorem 4.13], $\Sigma _i$ extends as a $C^{1,\alpha }$ free boundary hypersurface across p for some $\alpha \in (0,1)$. Higher regularity of $\Sigma _i$ follows from the prescribing mean curvature equation and elliptic regularity. $\square$

Appendix B: The second variation of $\mathcal {A}^h$ for smooth hypersurfaces

Our goal is to derive the second variation of ${\mathcal {A}}^h$ for arbitrary hypersurfaces with boundary in $(M^{n+1},\partial M,g)$. Note that we always assume $(M^{n+1},\partial M,g)$ is isometrically embedded in some closed $(n+1)$-dimensional Riemannian manifold $(\widetilde{M},\widetilde{g})\subset \mathbb {R}^L$. Let $(\Sigma ^n,\partial \Sigma )\subset (M,\partial M)$ be an embedded hypersurface in M with boundary on $\partial M$. We consider a two-parameter family of ambient variations $\Sigma (t,s)$ of $\Sigma = \Sigma (0, 0)$ defined by

$$\begin{aligned} \Sigma (t,s)= F_{t,s}(\Sigma ) = \Psi _s\circ \Phi _t(\Sigma ), \end{aligned}$$

where $\Phi _t$ and $\Psi _s$ are the flows generated by compactly supported vector fields X and Z in $\mathbb {R}^L$, respectively. We assume both $X,Z\in \widetilde{{\mathfrak {X}}}(M,\Sigma )$. Therefore each $\Sigma (t, s)$ is an embedded hypersurface in $\widetilde{M}$ with boundary lying on $\partial M$.

We have

$$\begin{aligned}\frac{\partial F_{t,s}}{\partial t}(x)\Big |_{t=s=0}=X(x),\ \frac{\partial F_{t,s}}{\partial s}(x)\Big |_{t=s=0}=Z(x),\ \text {and } \frac{\partial }{\partial t}\frac{\partial }{\partial s}F_{t,s}(x)\Big |_{t=s=0}=D_XZ(x). \end{aligned}$$

The computation in [3, Appendix A] gives that

$$\begin{aligned}&\frac{\partial }{\partial t}\frac{\partial }{\partial s}\Big |_{t=s=0}\mathcal {H}^n(\Sigma (t,s)) = \frac{\partial }{\partial t}\Big |_{t=0}\int _{\Sigma (t,0)}\textrm{div}Z\,d\mathcal {H}^n\nonumber \\&\quad =\int _{\Sigma }\Big ( \langle \nabla ^\perp (X^\perp ),\nabla ^\perp (Z^{\perp })\rangle -{{\,\textrm{Ric}\,}}(X^\perp ,Z^\perp )-|A|^2\langle X^\perp ,Z^\perp \rangle \Big ) \,d\mathcal {H}^n+\int _{\partial \Sigma }\langle \nabla _{X^\perp } Z^\perp ,\nu _{\partial M}\rangle \,d\mathcal {H}^{n-1} \nonumber \\&\qquad +\,\int _\Sigma \Xi _1(X,Z,\textbf{H})\, d\mathcal {H}^{n} + \int _{\partial \Sigma }\Xi _2(X,Z,\textbf{H},\varvec{\eta },\nu _{\partial M}) \, d\mathcal {H}^{n-1}, \end{aligned}$$

(B.1)

where

$$\begin{aligned} \Xi _1(X,Z,\textbf{H}) =\langle X^\perp ,\textbf{H}\rangle \langle Z^\perp ,\textbf{H}\rangle -\langle \nabla _{X^\perp }(Z^{\perp }),\textbf{H}\rangle -\langle [X^\perp ,Z^\top ],\textbf{H}\rangle , \end{aligned}$$

and

$$\begin{aligned} \Xi _2(X,Z,\textbf{H},\varvec{\eta },\nu _{\partial M})&=\langle \nabla _{X^\perp }Z^\perp ,\varvec{\eta }-\nu _{\partial M}\rangle +\langle [X^\perp ,Z^\top ],\varvec{\eta }\rangle +\textrm{div}_{\Sigma }(Z^\top )\langle X^\top ,\varvec{\eta }\rangle \\&\quad -\langle Z^\perp ,\textbf{H}\rangle \langle X^\top ,\varvec{\eta }\rangle -\langle X^\perp ,\textbf{H}\rangle \langle Z^\top ,\varvec{\eta }\rangle . \end{aligned}$$

Here $\textbf{H}=-H\nu$ is the mean curvature vector; $\varvec{\eta }$ is the unit co-normal vector field of $\Sigma$; $\nabla$ is the connection on M; $\perp$ and $\top$ are the normal and tangential parts on $\Sigma (t,s)$, respectively.

Let h be a smooth function on $\widetilde{M}$. Then a direct computation gives that

$$\begin{aligned}&\frac{\partial }{\partial t}\Big |_{t=0}\int _{\Sigma (t,0)}h\langle Z,\nu \rangle \,d\mathcal {H}^{n+1}\nonumber \\&\quad =\int _\Sigma \langle Z,h\nu \rangle \textrm{div} X+\nabla _X\langle Z,h\nu \rangle \, d\mathcal {H}^{n}\nonumber \\&\quad =\int _\Sigma \Big (\langle Z,h\nu \rangle \textrm{div}X^\top - \langle Z,h\nu \rangle \langle \textbf{H},X^\perp \rangle +\nabla _{X^\top }\langle Z,h\nu \rangle +(\nabla _{X^\perp }h)\langle Z,\nu \rangle +h(\nabla _{X^\perp }\langle Z,\nu \rangle )\Big )\,d\mathcal {H}^n\nonumber \\&\quad =\int _{\Sigma }\Big ((\partial _\nu h)\langle X^\perp ,Z^\perp \rangle -\langle X,\textbf{H}\rangle \langle Z,h\nu \rangle + \langle \nabla _{X^\perp }(Z^\perp ),h\nu \rangle \Big )\,d\mathcal {H}^n+\int _{\partial \Sigma }\langle Z^\perp ,h\nu \rangle \langle X^\top ,\varvec{\eta }\rangle \,d\mathcal {H}^{n-1}. \end{aligned}$$

(B.2)

By (B.1) and (B.2), we have

$$\begin{aligned}&\frac{\partial }{\partial t}\Big |_{t=0} \int _{\Sigma (t,0)}\textrm{div}Z-h\langle Z,\nu \rangle \, d\mathcal {H}^n\\&\quad =\int _{\Sigma }\Big ( \langle \nabla ^\perp (X^\perp ),\nabla ^\perp (Z^{\perp })\rangle -{{\,\textrm{Ric}\,}}(X^\perp ,Z^\perp ) -|A|^2\langle X^\perp ,Z^\perp \rangle -(\partial _\nu h)\langle X^\perp ,Z^\perp \rangle \Big )\,d\mathcal {H}^n\\&\qquad +\int _{\partial \Sigma }\langle \nabla _{X^\perp }Z^\perp ,\nu _{\partial M} \rangle \,d\mathcal {H}^{n-1} +\int _\Sigma \widetilde{\Xi }_1(X,Z,\textbf{H}) \, d\mathcal {H}^{n}+\int _{\partial \Sigma }\widetilde{\Xi }_2(X,Z,\textbf{H},\varvec{\eta },\nu _{\partial M})\, d\mathcal {H}^{n-1}, \end{aligned}$$

where

$$\begin{aligned} \widetilde{\Xi }_1(X,Z,\textbf{H})&=\Xi _1(X,Z,\textbf{H})+\langle X^\perp ,\textbf{H}\rangle \langle Z^\perp ,h\nu \rangle - \langle \nabla _{X^\perp }(Z^\perp ),h\nu \rangle \\&=\langle X^\perp ,\textbf{H}\rangle \langle Z^\perp ,\textbf{H}+h\nu \rangle -\langle \nabla _{X^\perp }(Z^{\perp }),\textbf{H}+h\nu \rangle -\langle [X^\perp ,Z^\top ],\textbf{H}\rangle \\&=\langle X^\perp ,\textbf{H}\rangle \langle Z^\perp ,\textbf{H}+h\nu \rangle -\langle \nabla _{X^\perp }(Z^{\perp }),\textbf{H}+h\nu \rangle -A(X^\top ,Z^\top )\langle \nu ,\textbf{H}\rangle , \end{aligned}$$

and

$$\begin{aligned} \widetilde{\Xi }_2(X,Z,\textbf{H},\varvec{\eta },\nu _{\partial M})&=\Xi _2(X,Z,\textbf{H},\varvec{\eta },\nu _{\partial M})- \langle Z^\perp ,h\nu \rangle \langle X^\top ,\varvec{\eta }\rangle \\&=\langle \nabla _{X^\perp }Z^\perp ,\varvec{\eta }-\nu _{\partial M}\rangle +\langle [X^\perp ,Z^\top ],\varvec{\eta }\rangle +\textrm{div}_{\Sigma }(Z^\top )\langle X^\top ,\varvec{\eta }\rangle \\&\quad -\langle Z^\perp ,\textbf{H}+h\nu \rangle \langle X^\top ,\varvec{\eta }\rangle -\langle X^\perp ,\textbf{H}\rangle \langle Z^\top ,\varvec{\eta }\rangle . \end{aligned}$$

In the equality of $\widetilde{\Xi }_1(X,Z,\textbf{H})$, we used the following identity:

$$\begin{aligned} \langle [X^\perp ,Z^\top ],\nu \rangle&=\langle \nabla _{X^\perp }(Z^\top )-\nabla _{Z^\top }(X^\perp ),\nu \rangle \\&=\langle \nabla _{X}(Z^\top )-\nabla _{X^\top }(Z^\top )-\nabla _{Z^\top }(X^\perp ),\nu \rangle \\&=-\langle Z^\top ,\nabla _{X}\nu \rangle -\langle \nabla _{Z^\top }(X^\perp ),\nu \rangle +A(X^\top ,Z^\top )\\&=\langle Z^\top ,\nabla \langle X,\nu \rangle \rangle -\langle \nabla _{Z^\top }(X^\perp ),\nu \rangle +A(X^\top ,Z^\top )\\&= A(X^\top ,Z^\top ). \end{aligned}$$

We remark that

$$\begin{aligned} |\widetilde{\Xi }_1(X,Z,\textbf{H})|+|\widetilde{\Xi }_2(X,Z,\textbf{H},\varvec{\eta },\nu _{\partial M})| \le C\big (|X|(|\textbf{H}+h\nu |+| \varvec{\eta }-\nu _{\partial M} |+|Z^\top |)+|X^\top |\big ). \end{aligned}$$

Appendix C: Cut-off trick

In this section, we provide a lemma that has been used in Part 3 of the proof of Theorem 2.9 (v). Such a result has also been used in [20].

Lemma C.1

Let $(M^{n+1},\partial M,g)$ be a compact manifold with boundary of dimension $(n+1)\ge 3$. Let $(\Sigma ,\partial \Sigma )\subset (M,\partial M)$ be an almost embedded free boundary h-hypersurface and $\varphi$ be a smooth function on $\Sigma$. Then for any $p\in \Sigma$, there exists a family of cut-off functions $(\xi _r)_{0<r<\epsilon }$ for some $\epsilon >0$ so that $\xi _r(p)=0$

$$\begin{aligned} \textrm{II}_\Sigma (\varphi ,\varphi )=\lim _{r\rightarrow 0}\textrm{II}_\Sigma (\xi _r\varphi ,\xi _r\varphi ). \end{aligned}$$

Proof

Set

$$\begin{aligned} \xi _r(x)=\left\{ \begin{aligned}&0,&|x|<r^2\\&2-\frac{\log |x|}{\log r},&r^2\le |x|\le r\\&1,&|x|>r \end{aligned}\right. , \end{aligned}$$

(C.1)

where $|x|={\text {dist}}_M(x,p)$. Then we have $\int _\Sigma |\nabla \xi _r|^2<C(n)/|\log r|\rightarrow 0$ and $\xi _r\rightarrow 1$ as $r\rightarrow 0$. Then it suffices to prove $\int _{\Sigma }|\nabla (\xi _r\varphi )|^2\rightarrow \int _{\Sigma }|\nabla \varphi |^2$ as $r\rightarrow 0$. This follows from $\int _\Sigma |\nabla \xi _r|^2\rightarrow 0$ $\square$

Appendix D: Local h-foliation with free boundary

The following proposition is a generalization of minimal foliation given by White [55]. This description has already been stated in [54, Proposition A.2].

Proposition D.1

Let $(M^{n+1},\partial M,g)$ be a compact Riemannian manifold with boundary, and let $(\Sigma ,\partial \Sigma )\subset (M,\partial M)$ be an embedded, free boundary minimal hypersurface. Given a point $p\in \partial \Sigma$, there exist $\epsilon >0$ and a neighborhood $U\subset M$ of p such that if $h:U\rightarrow \mathbb {R}$ is a smooth function with $\Vert h\Vert _{C^{2,\alpha }}<\epsilon$ and

$$\begin{aligned} w:\Sigma \cap U\rightarrow \mathbb {R} \text { satisfies } \Vert w\Vert _{C^{2,\alpha }}<\epsilon , \end{aligned}$$

then for any $t\in (-\epsilon ,\epsilon )$, there exists a $C^{2,\alpha }$-function $v_t: U\cap \Sigma \rightarrow \mathbb {R}$, whose graph $G_t$ meets $\partial M$ orthogonally along $U\cap \partial \Sigma$ and satisfies:

$$\begin{aligned} H_{G_t} = h|_{G_t}, \end{aligned}$$

(where $H_{G_t}$ is evaluated with respect to the upward pointing normal of $G_t$), and

$$\begin{aligned} v_t(x) = w(x) + t, \text { if } x\in \partial (U\cap \Sigma )\cap \textrm{Int} M. \end{aligned}$$

Furthermore, $v_t$ depends on t, h, w in $C^1$ and the graphs $\{G_t: t\in [-\epsilon , \epsilon ]\}$ forms a foliation.

Proof

The proof follows from [55, Appendix] together with the free boundary version [3, Section 3]. The only modification is that we need to use the following map to replace $\Phi$ in [3, Section 3]:

$$\begin{aligned} \Psi : \mathbb {R} \times X \times Y\times Y \times Y \rightarrow Z_1 \times Z_2 \times Z_3. \end{aligned}$$

The map $\Psi$ is defined by

$$\begin{aligned} \Psi (t,g,h,w,u) = (H_{g(t+ w + u)}-h, g(N_g(t + w + u), \nu _g (t + w + u)), u|_{\Gamma _2} ); \end{aligned}$$

here all the notions are the same as [3, Section 3]. $\square$