An improved eigenvalue estimate for embedded minimal hypersurfaces in the sphere

An important problem in geometric analysis is to understand the spectrum of the Laplace-Beltrami operator on a Riemannian manifold, and to study its relation to the underlying intrinsic and/or ambient geometry. From the geometric perspective, it is of particular interest to address such questions for manifolds embedded in spaces of constant curvature. In this paper, we obtain a new lower bound for the first non-zero eigenvalue $\lambda_{1}(\Sigma)$ of the induced Laplace-Beltrami operator $-\Delta^{\Sigma}$ on a smooth closed hypersurface $\Sigma^{n}$ minimally embedded in the unit sphere $\mathbb{S}^{n+1}$ (which we always assume to be equipped with the round metric).

In this direction, an argument of Choi & Wang [6], later refined by Brendle [1], gives the lower bound

An important application of (1.1) and the Yang-Yau inequality [16] is an area bound for embedded minimal surfaces in $\mathbb{S}^{3}$ in terms of their genus; this plays a crucial role in the compactness theory of Choi & Schoen [5]. Moreover, (1.1) provides evidence towards a famous conjecture of Yau [17], which predicts that $\lambda_{1}(\Sigma)$ is equal to $n$. Note that the restriction to $\mathbb{S}^{n+1}$ of each coordinate function on $\mathbb{R}^{n+2}$ is an eigenfunction for $-\Delta^{\Sigma}$ with eigenvalue $n$, and thus the upper bound $\lambda_{1}(\Sigma)\leq n$ is clear.

Despite an extensive literature relating to the study of $\lambda_{1}(\Sigma)$ under additional assumptions on $\Sigma$ since the work of Choi & Wang (see e.g. [4, 15] and the references therein), (1.1) has remained the strongest explicit lower bound that is known to hold for a general embedded minimal hypersurface in $\mathbb{S}^{n+1}$. In this paper, we obtain an explicit improvement on (1.1) which depends only on the dimension $n$ and an upper bound on the length $\|A\|\mathrel{\hbox to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\sqrt{\operatorname{trace}A^{2}}$ of the second fundamental form of $\Sigma$. Our main result is as follows:

Whilst we are only interested in explicitly computable lower bounds for $\lambda_{1}(\Sigma)$ in this paper, we note that upper bounds for either $\lambda_{1}(M^{n},g)$ or $\lambda_{1}(M^{n},g)\cdot\operatorname{Vol}(M^{n},g)^{2/n}$ on Riemannian manifolds $(M^{n},g)$ have also been studied extensively – see for instance the classical works of Cheng [3], Li & Yau [10, 11], Yang & Yau [16] and Korevaar [9]. In particular, recall that for a closed orientable Riemannian surface $(\Sigma^{2},g)$ of genus $\gamma$, the Yang-Yau inequality [16, 7] states that $\lambda_{1}(\Sigma,g)\operatorname{Area}(\Sigma,g)\leq 8\pi\lfloor\frac{\gamma+3}{2}\rfloor$, where $\lfloor x\rfloor$ denotes the integer part of $x$. The following result is then an immediate corollary of the Yang-Yau inequality and Theorem 1.1:

To put Theorem 1.1 into context, we now briefly discuss some related results. We first observe that, in light of the strictness of the inequality in (1.1), non-explicit improved lower bounds depending only on quantities such as dimension, index, topological type and curvature bounds follow from suitable compactness results. For example, if $\mathcal{A}(\Lambda,n)$ denotes the class of closed embedded minimal hypersurfaces in $\mathbb{S}^{n+1}$ satisfying $\max_{\Sigma}\|A\|\leq\Lambda$, then it is well-known that $\mathcal{A}(\Lambda,n)$ is compact in the $C^{k}$ topology for any $k\geq 2$. Combined with (1.1), it follows that there exists a constant $\alpha(\Lambda,n)>0$ such that

We stress that, in contrast with (1.6), the estimate (1.3) obtained in Theorem 1.1 provides an explicitly computable improvement on (1.1). Moreover, our lower bound (1.3) is obtained by arguing more directly in the spirit of [6], rather than appealing to any compactness theory.

$C^{k}$ compactness results have also been established in other classes. For example, Choi & Schoen showed in [5] that the class $\mathcal{B}(\gamma)$ of closed embedded minimal surfaces in $\mathbb{S}^{3}$ with genus less than $\gamma$ is compact in the $C^{k}$ topology for any $k\geq 2$. In combination with (1.1), this implies the existence of a constant $\beta(\gamma)>0$ such that

A more recent compactness of result of Sharp [13, Corollary 2.6] shows that the class $\mathcal{C}(V,I,n)$ of closed embedded minimal hypersurfaces in $\mathbb{S}^{n+1}$ with volume bounded from above by $V$ and index bounded from above by $I$ is compact in the $C^{k}$ topology for $k\geq 2$ when $2\leq n\leq 6$. Combined with (1.1), this implies the existence of a constant $\delta(V,I,n)>0$ such that

In a similar vein to Theorem 1.1, it would be interesting to derive improved lower bounds for $\lambda_{1}(\Sigma)$ with explicit dependence on quantities such as genus (when $n=2$), volume and/or index. Such results could provide a step towards proving Yau’s conjecture within certain classes of minimal hypersurfaces in $\mathbb{S}^{n+1}$. Recently, Yau’s conjecture was established for the class of embedded isoparametric minimal hypersurfaces in $\mathbb{S}^{n+1}$ – see Tang & Yan [15] and the references therein. We refer also to the work of Choe & Soret [4], where Yau’s conjecture was established for a class of symmetric minimal surfaces in $\mathbb{S}^{3}$.

The plan of the paper is as follows. In Section 2 we prove a preliminary result on the embeddedness of parallel hypersurfaces in $\mathbb{S}^{n+1}$. As a corollary, we obtain an explicit volume bound for closed embedded mean-convex hypersurfaces in $\mathbb{S}^{n+1}$ in terms of an upper bound for $\|A\|$. In Section 3 we prove Theorem 1.1. The key here is to estimate a positive term which is dropped in the estimate of Choi & Wang in [6]. Here, our integral estimates require working in a neighbourhood of $\Sigma$ whose thickness is controlled away from zero; this control is provided by our results in Section 2.

Suppose that $\Sigma^{n}$ is a smooth, closed and embedded hypersurface in $\mathbb{S}^{n+1}$. As observed in [6], $\Sigma$ divides the sphere into two components $\mathbb{S}^{n+1}=M_{1}\cup M_{2}$, where $\partial M_{1}=\partial M_{2}=\Sigma$. Let $N\Sigma$ denote the normal bundle of $\Sigma\subset\mathbb{S}^{n+1}$ and $\operatorname{exp}^{N\Sigma}$ the corresponding exponential map. We fix the orientation on $\Sigma$ determined by the unit normal vector field $X$ on $\Sigma$ pointing into $M_{1}$, and for $t\in\mathbb{R}$ we define

Geometrically, $\Sigma^{t}$ is the hypersurface parallel to $\Sigma$ and of signed distance $t$ to $\Sigma$. It is well-known (see e.g. Theorems 2.1 and 2.2 in [2]) that if $\kappa_{1}(p),\dots,\kappa_{n}(p)$ are the principal curvatures of $\Sigma$ at $p$ and $\kappa_{\operatorname{max}}=\max_{p\in\Sigma,i\in\{1,\dots,n\}}|\kappa_{i}(p)|$, then $\Sigma^{t}$ is a smooth immersed hypersurface in $\mathbb{S}^{n+1}$ for

Moreover, we may consider $n$ continuous functions $\kappa_{i}(\cdot,\cdot):\Sigma\times(-T_{\Sigma},T_{\Sigma})\rightarrow\mathbb{R}$ defined by

Then for each $t\in(-T_{\Sigma},T_{\Sigma})$, the quantities $\kappa_{1}(p,t),\dots,\kappa_{n}(p,t)$ are the principal curvatures of $\Sigma^{t}$ at $\operatorname{exp}^{N\Sigma}(p,tX_{p})$, with respect to the orientation on $\Sigma^{t}$ determined by parallel transporting $X$ along geodesics normal to $\Sigma$ by a signed distance $t$.

Whilst it is well-known that $\Sigma^{t}$ remains embedded for $t$ sufficiently small, in general the range of $t$ for which $\Sigma^{t}$ is embedded is not controlled by the curvature of $\Sigma$, since $\Sigma$ may be arbitrarily close to ‘self-touching’. We show that in the case that $\Sigma$ is mean-convex (that is, the mean curvature $H_{\Sigma}$ of $\Sigma$ is nonnegative), we do in fact have such control:

As a corollary of Proposition 2.1 we obtain an explicit volume bound for closed embedded mean-convex hypersurfaces in $\mathbb{S}^{n+1}$:

In this section we prove Theorem 1.1. We begin in Section 3.1 by recalling the argument of Choi & Wang [6]. In Section 3.2 we give the proof of Theorem 1.1 assuming the validity of two propositions. In Sections 3.3 and 3.4 we give the proofs of these two propositions.