Generic scarring for minimal hypersurfaces along stable hypersurfaces

Abstract

Let $M^{n+1}$ be a closed manifold of dimension $3\le n+1\le 7$. We show that for a $C^\infty$-generic metric g on M, to any connected, closed, embedded, 2-sided, stable, minimal hypersurface $S\subset (M,g)$ corresponds a sequence of closed, embedded, minimal hypersurfaces $\{\Sigma _k\}$ scarring along S, in the sense that the area and Morse index of $\Sigma _k$ both diverge to infinity and, when properly renormalized, $\Sigma _k$ converges to S as varifolds. We also show that scarring of immersed minimal surfaces along stable surfaces occurs in most closed Riemannian 3-manifods.

Appendix A. Differentiability of the first eigenvalue

For the reader’s convenience, we give a proof of the known fact that the first eigenvalue of a self-adjoint elliptic operator depends in a differentiable way on the coefficients; (see [Uhl76] for related results). Let $\Sigma$ be a smooth closed n-manifold. Fix two integers $r\ge 2$, $k\ge 1$. Let U be a smooth Banach manifold and let $\Phi {:}\, U \rightarrow \big (C^r(\Sigma )\big )^{n^2+n+1}$ be a $C^k$ map which associates to any $\gamma \in U$ a triple $\Phi (\gamma ) = \big ((a^{ij})_{1\le i, j \le n}, (b^i)_{1\le i\le n}, c\big )$, where $a^{ij}, b^i, c\in C^r(\Sigma )$, such that $(a^{ij})>0$ is positive definite and such that the following elliptic operator

$$\begin{aligned} L_{\gamma }u := a^{ij}u_{ij} + b^i u_i + cu \end{aligned}$$

is self-adjoint with respect to a volume measure depending on $\gamma \in U$.

A number $\lambda _1(\gamma )$ is the first eigenvalue of $L_{\gamma }$ if and only if

$$\begin{aligned} L_{\gamma } \varphi _1 = -\lambda _1(\gamma ) \varphi _1 \end{aligned}$$

for some $\varphi _1 \in H^{1}(\Sigma )\cap C^0(\Sigma )$ with $\varphi _1>0$. The eigenfunctions of $L_{\gamma }$ span $L^2(\Sigma )$ and are in $C^r(\Sigma )$.

Lemma A.1

$\lambda _1(\gamma )$ is a $C^k$ function of $\gamma \in U$.

Proof

Fix a metric $g_0$ on $\Sigma$ and let $S^{r}(\Sigma )$ be the space of functions $u\in H^{r}(\Sigma )$ with $\Vert u\Vert _{L^2(\Sigma ,g_0)}=1$, where the $L^2$-norm is computed with $g_0$. Note that $H^{r}(\Sigma )$ does not depend on the metric. Consider the operator $T{:}\, U\times \mathbb {R}\times S^{r}(\Sigma ) \rightarrow H^{r-2}(\Sigma )$ defined by

$$\begin{aligned} T\big (\gamma , \mu , u\big ) = L_{\gamma }u + \mu u. \end{aligned}$$

Since $\Phi {:}\,U \rightarrow \big (C^r(\Sigma )\big )^{n^2+n+1}$ is a $C^k$ map by assumption, T is also $C^k$-differentiable in its variables. We have $T\big (\gamma , \lambda , \varphi \big )=0$ if and only if $\lambda$ is an eigenvalue of $L_{\gamma }$ and $\varphi$ is an associated normalized eigenfunction. An eigenvalue is simple if the associated eigenspace is one dimensional. It is known that the first eigenvalue of $L_{\gamma }$ is always simple. Given $\gamma _1 \in U$, consider the first eigenvalue $\lambda _1$ of $L_{\gamma _1}$ and a normalized positive first eigenfunction $\varphi _1\in S^{r}(\Sigma )$, and then consider the following differential

$$\begin{aligned} D_{(\mu , u)} T|_{(\gamma _1, \lambda _1, \varphi _1)}(s, v) = L_{\gamma _1}v + \lambda _1 v + s \varphi _1, \end{aligned}$$

where $s\in \mathbb R$, $v\in \text {span}\{\varphi _1\}^{\perp _{g_0}} = \{ u\in H^{r}(\Sigma ), \int _{\Sigma } u \varphi _1d\text {vol}_{g_0}=0\}$. We know that the null space of $L_{\gamma _1}+\lambda _1$ is $\text {span}\{\varphi _1\}$. Moreover since $L_{\gamma _1} +\lambda _1$ is self adjoint for a volume measure $\nu$, and since $\lambda _1$ is simple, the image of $\text {span}\{\varphi _1\}^{\perp _{g_0}}$ by this operator is $\{ u\in H^{r-2}(\Sigma ), \int _{\Sigma } u \varphi _1 d\nu =0\}$. Hence

$$\begin{aligned} D_{(\mu , u)} T|_{(\gamma _1, \lambda _1, \varphi _1)}: \mathbb R\times \text {span}\{\varphi _1\}^{\perp _{g_0}} \rightarrow H^{r-2}(\Sigma ) \end{aligned}$$

is an isomorphism. By the Implicit Function Theorem, near $\gamma _1$, there exist $C^k$ maps:

$$\begin{aligned} \gamma \rightarrow \lambda (\gamma )\in \mathbb {R}\quad \text {and} \quad \gamma \rightarrow \varphi (\gamma ) \in H^{r}(\Sigma ) \end{aligned}$$

with

$$\begin{aligned} \lambda (\gamma _1)=\lambda _1 \quad \text {and} \quad \varphi (\gamma _1)= \varphi _1, \end{aligned}$$

such that $L_{\gamma }\varphi (\gamma ) = -\lambda (\gamma )\varphi (\gamma )$. Note that $\varphi (\gamma )$ is also positive when $\gamma$ is close enough to $\gamma _1$, so $\lambda (\gamma )$ is indeed the first eigenvalue of $L_\gamma$. The conclusion then follows.$\square$