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 $\varphi$ into ${\mathrm{\S }}^{n+1}$. Let $q>n$ and $A>0$ be some real numbers satisfying $\mathrm{\mid }M{\mathrm{\mid }}^{\frac{1}{n}}(1+\mathrm{\parallel }B{\mathrm{\parallel }}_q)\le A$. Suppose that $\varphi (M)\subset B(p_0,R)$, where $p_0$ is a center of gravity of $M$ and radius $R<\frac{\pi }{2}$. We prove that there exists a positive constant $\text{e}$ depending on $q$, $n$, $R$ and $A$ such that if $n(1+\mathrm{\parallel }H{\mathrm{\parallel }}_{\mathrm{\infty }}^2)-\text{e}\le {\text{l}}_1$, then $M$ is diffeomorphic to ${\mathrm{\S }}^n$. Furthermore, $\varphi (M)$ is starshaped with respect to $p_0$, Hausdorff close and almost-isometric to the geodesic sphere S\(p_0,R_0\), where $R_0=\arcsin \frac{1}{\sqrt{1+\mathrm{\parallel }H{\mathrm{\parallel }}_{\mathrm{\infty }}^2}}$.