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 (Mⁿ, g) be a closed, connected, and oriented Riemannian manifold isometrically immersed by ϕ into S^{n + 1}. Let q> n and A> 0 be some real numbers satisfying |M|1n(1+‖B‖q)≤A. Suppose that ϕ(M) ⊂ B¯ (p₀, R) , where p₀ is a center of gravity of M and radius R< π/ 2. We prove that there exists a positive constant ε depending on q, n, R, and A such that if n(1+‖H‖∞2)-ε≤λ1, then M is diffeomorphic to Sⁿ. Furthermore, ϕ(M) is starshaped with respect to p₀, almost-isometric to the geodesic sphere S(p₀, R₀) , where R0=arcsin11+‖H‖∞2.