An Eigenvalue Pinching Theorem for Compact Hypersurfaces in A Sphere

TL;DR

This paper establishes a theorem linking the first Laplacian eigenvalue to the geometry of compact hypersurfaces in a sphere, showing near-eigenvalue conditions imply the hypersurface is close to a geodesic sphere.

Contribution

It proves an eigenvalue pinching theorem for hypersurfaces in spheres, connecting eigenvalues with geometric shape and topology, extending previous spectral geometry results.

Findings

Hypersurfaces with eigenvalues close to a specific bound are diffeomorphic to spheres.

Such hypersurfaces are Hausdorff close and almost-isometric to geodesic spheres.

The hypersurfaces are starshaped with respect to a point in the sphere.

Abstract

In this article, we prove an eigenvalue pinching theorem for the first eigenvalue of the Laplacian on compact hypersurfaces in a sphere. Let $(M^n,g)$ be a closed, connected and oriented Riemannian manifold isometrically immersed by $\phi$ into $\S^{n+1}$. Let $q>n$ and $A>0$ be some real numbers satisfying $|M|^\frac{1}{n}(1+\|B\|_q)\leq A$. Suppose that $\phi(M)\subset B(p_0,R)$, where $p_0$ is a center of gravity of $M$ and radius $R<\frac{\pi}{2}$. We prove that there exists a positive constant $\e$ depending on $q$, $n$, $R$ and $A$ such that if $n(1+\|H\|_\infty^2)-\e\leq \l_1$, then $M$ is diffeomorphic to $\S^n$. Furthermore, $\phi(M)$ is starshaped with respect to $p_0$, Hausdorff close and almost-isometric to the geodesic sphere $S\(p_0,R_0\)$, where $R_0=\arcsin\frac{1}{\sqrt{1+\|H\|_\infty^2}}$.