An Eigenvalue Pinching Theorem for Compact Hypersurfaces in A Sphere

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 $R_0=\arcsin\frac{1}{\sqrt{1+\|H\|_\infty^2}}$.