First stability eigenvalue of singular minimal hypersurfaces in spheres

Abstract

In this short note we extend an estimate due to J. Simons on the first stability eigenvalue of minimal hypersurfaces in spheres to the singular setting. Specifically, we show that any singular minimal hypersurface in ${\mathbf {S}}^{n+1}$, which is not totally geodesic and satisfies the $\alpha$-structural hypothesis, has first stability eigenvalue at most $-\,2n$, with equality if and only if it is a product of two round spheres. The equality case was settled independently in the classical setting by Wu and Perdomo.

Appendix A: A smooth cutoff construction

In this “Appendix” we reproduce the cutoff construction of Morgan and Ritoré [5]. We do so only for the case of submanifolds in Euclidean space with compact support and singular set of vanishing codimension 2 measure. The other cases considered in [5] are similar; see also Remark A.3. In this section only, B(x, r) denotes a Euclidean ball of radius r centred at x.

First we need an easy bound for the number of intersections of balls of comparable radii.

Lemma A.1

Let ${\mathcal {B}}=\{B(p_i,r_i)\}$ be a collection of balls in ${\mathbb {R}}^N$. Suppose that there are $\alpha ,\beta \ge 1$ such that the sub-balls $\{B(p_i,r_i/\alpha )\}$ are pairwise disjoint, and that the radii are $\beta$-comparable, that is, $\sup r_i \le \beta \inf r_i$. Then each ball in ${\mathcal {B}}$ intersects at most $(3\alpha \beta )^N-1$ other balls in ${\mathcal {B}}$.

Proof

Fix a ball $B(p_i,r_i) \in {\mathcal {B}}$. Any ball in ${\mathcal {B}}$ that intersects $B(p_i,r_i)$ must be contained in the larger ball $B(p_i, r_i + 2\sup r_j)$, which has volume at most $\omega _N (3\sup r_j)^N$. Here $\omega _N$ is the volume of the unit ball in ${\mathbb {R}}^N$.

On the other hand, since the sub-balls $\{B(p_j,r_j/\alpha )\}$ are pairwise disjoint, each must take up a volume at least $\omega _N (\inf r_j/\alpha )^N$ in the larger ball $B(p_i, r_i + 2\sup r_j)$. So at most $\frac{(3\sup r_j)^N}{(\inf r_j/\alpha )^N} \le (3\alpha \beta )^N$ sub-balls can fit in the larger ball, and this implies the result. $\square$

We now proceed to the construction of smooth cutoff functions around the singular set.

Proposition A.2

( [5]) Let $\Sigma ^k$ be a smooth embedded submanifold in ${\mathbb {R}}^N$ with bounded mean curvature and compact closure $\overline{\Sigma }$. If ${\text {sing}}\Sigma = \overline{\Sigma }{\setminus }\Sigma$ satisfies ${\mathcal {H}}^{k-2}({\text {sing}}\Sigma )=0$, then for any $\epsilon >0$ there exists a smooth function $\varphi _\epsilon : \overline{\Sigma } \rightarrow [0,1]$ supported in $\Sigma$ such that:

• (1) ${\mathcal {H}}^k(\{\varphi _\epsilon \ne 1\}) <\epsilon$;

• (2) $\int _\Sigma |\nabla \varphi _\epsilon |^2 < \epsilon$;

• (3) $\int _\Sigma |\Delta \varphi _\epsilon | < \epsilon$.

Proof

Fix a smooth radial cutoff function $\varphi :{\mathbb {R}}^N\rightarrow [0,1]$ such that $\phi =0$ in B(0, 1 / 2) and $\phi =1$ outside B(0, 1). The derivatives are bounded, say $|D\varphi |^2 + |D^2 \varphi | \le C_0$. By scaling $\varphi$ to B(x, r), $r\le 1$, we get a cutoff function satisfying $|D\varphi |^2 + |D^2 \varphi | \le C_0 r^{-2}$.

Since $\Sigma$ has bounded mean curvature $|\vec {H}|\le C_H$, the monotonicity formula [8] implies that there is a constant $C_V$ such that ${\mathcal {H}}^k(\Sigma \cap B(x,r)) \le C_V r^k$ for any $r\le 1$ and any x. Moreover, on $\Sigma \cap B_r(x)$ we will have

$$\begin{aligned} |\Delta \varphi | \le k|D^2\varphi | + |\langle \vec {H}, D\varphi \rangle | \le kC_0 r^{-2} + C_H \sqrt{C_0} r^{-1} \le C_1 r^{-2}. \end{aligned}$$

(A.1)

Now let $\epsilon >0$. By definition of Hausdorff measure we may cover the singular set by finitely many balls $\{B(p_i,r_i/6)\}$ such that $r_i \le 1$ and $\sum _{i} r_i^{k-2} <\epsilon$. Consider the enlarged cover $\{B(p_i,r_i/2)\}$. If $B(p_i,r_i/6)\cap B(p_j,r_j/6)\ne \emptyset$, $r_i \ge r_j$, then certainly $B(p_j, r_j/6) \subset B(p_i,r_i/2)$, so we could discard any such j. In doing so we obtain a cover $\{B(p_i,r_i/2)\}$ such that the $\{B(p_i,r_i/6)\}$ are pairwise disjoint. We may relabel the radii so that $r_1\ge r_2\ge \cdots$, and we partition the balls into classes of comparable radii ${\mathcal {B}}_m = \{i | 2^m \le r_i < 2^{m+1}\}$.

Cut off on each $B(p_i,r_i)$ by scaled cutoff functions $\varphi _i$ as above, then set $\varphi _\epsilon = \prod _i \varphi _i$. Immediately we have ${\mathcal {H}}^k(\{\varphi _\epsilon \ne 1\}) \le \sum _i C_V r_i^{n} < C_V \epsilon$. For properties (2) and (3) we must bound the sum of all product terms $\int _\Sigma |\nabla \varphi _i||\nabla \varphi _j|$. Such a term is zero if $B(p_i,r_i)$ and $B(p_j,r_j)$ are disjoint, otherwise we have the bound

$$\begin{aligned} \int _\Sigma |\nabla \varphi _i| |\nabla \varphi _j| \le \frac{C_0 C_V}{r_i r_j} \min (r_i,r_j)^k. \end{aligned}$$

(A.2)

The procedure to estimate these cross terms is as follows: We fix j and consider the sum over $i\le j$ (that is, $r_i \ge r_j$). Letting ${\mathcal {B}}_{m_j}$ be the class containing j, we will bound the number of intersections that $B(p_j,r_j)$ can have with balls $B(p_i,r_i)$ in each class ${\mathcal {B}}_{m_j +h}$, $h\ge 0$.

The key observation is that if $B(p_i,r_i) \cap B(p_j,r_j) \ne \emptyset$, then certainly the enlarged ball $B(p_i, r_i+r_j)$ must contain the point $p_j$. In particular all such enlarged balls must intersect each other. But for $i\in {\mathcal {B}}_{m_j+h}$, the radii $r_i+r_j$ are comparable to within a factor of

$$\begin{aligned} \frac{2^{m_j+h+1} + r_j}{2^{m_j+h}+r_j} \le \frac{2^{m_j+h+1} + 2^{m_j+1}}{2^{m_j+h}+2^{m_j}} = 2. \end{aligned}$$

(A.3)

For any such i we also have $\frac{r_j}{r_i} \le \frac{2^{m_j+1}}{2^{m_j+h}} = 2^{1-h},$ so in particular $\frac{r_i+r_j}{r_i} \le 3$ and the sub-balls $\{B(p_i,\frac{r_i+r_j}{18}) | i \in {\mathcal {B}}_{m_j+h}\}$ must be pairwise disjoint. Thus by Lemma A.1,

$$\begin{aligned} \#\{i \in {\mathcal {B}}_{m_j+h} | B(p_i,r_i) \cap B(p_j,r_j) \ne \emptyset \} \le \#\{ i \in {\mathcal {B}}_{m_j+h} | B(p_i, r_i+r_j) \ni p_j\} \le 108^N.\nonumber \\ \end{aligned}$$

(A.4)

Using again that $\frac{r_j}{r_i} \le 2^{1-h}$ for $i\in {\mathcal {B}}_{m_j+h}$, the estimate (A.2) gives

$$\begin{aligned} \int _\Sigma |\nabla \varphi _i| |\nabla \varphi _j| \le C_0 C_V r_j^{k-1} r_i^{-1} \le C_0 C_V 2^{1-h} r_j^{k-2}. \end{aligned}$$

(A.5)

Then by (A.4) we have that

$$\begin{aligned} \sum _{\begin{array}{c} i\le j\\ i \in {\mathcal {B}}_{m_j+h} \end{array}} \int _\Sigma |\nabla \varphi _i| |\nabla \varphi _j| \le 108^N C_0 C_V 2^{1-h} r_j^{k-2}, \end{aligned}$$

(A.6)

and summing over $h\ge 0$ we get that $\sum _{i\le j} \int _\Sigma |\nabla \varphi _i| |\nabla \varphi _j| \le 4(108^N)C_0 C_V r_j^{k-2}.$

Finally, summing over j gives

$$\begin{aligned} \sum _{i,j} \int _\Sigma |\nabla \varphi _i| |\nabla \varphi _j|= & {} 2\sum _j \sum _{i\le j} \int _\Sigma |\nabla \varphi _i| |\nabla \varphi _j| \nonumber \\\le & {} 8(108^N)C_0 C_V \sum _j r_j^{k-2} < 8(108^N)C_0 C_V\epsilon . \end{aligned}$$

(A.7)

Since each $\varphi _j^2\le 1$ we conclude that

$$\begin{aligned} \int _\Sigma |\nabla \varphi _\epsilon |^2 \le \sum _{i,j} \int _\Sigma |\nabla \varphi _i| |\nabla \varphi _j| <8(108^N)C_0 C_V\epsilon \end{aligned}$$

(A.8)

and

$$\begin{aligned} \int _\Sigma |\Delta \varphi _\epsilon |\le & {} \sum _i \int _\Sigma |\Delta \varphi _i| + \sum _{i,j} \int _\Sigma |\nabla \varphi _i| |\nabla \varphi _j| \nonumber \\< & {} \sum _i C_1C_V r_i^{k-2} +8(108^N)C_0 C_V\epsilon < (C_1+8(108^N)C_0)C_V\epsilon . \end{aligned}$$

(A.9)

Since $\epsilon$ was arbitrary this concludes the proof. $\square$

Remark A.3

Morgan and Ritoré [5] state their construction also for somewhat more general assumptions as follows. If the ambient space is a regular cone, the argument proceeds with one extra cutoff around the vertex. If $k=2$ and the surface has isolated singular points, one may use logarithmic cutoff functions (and one does not need to be concerned with the intersections). Finally, to handle the noncompact case one may proceed by covering the singular set in annuli $B_{m+1}{\setminus } B_{m-1}$ with balls of radius $r_{m,i}< 1$ such that $\sum _i r_{m,i}^{k-2} \le \frac{2^{-n}\epsilon }{C_V(m)}$.