An Eigenvalue Pinching Theorem for Compact Hypersurfaces in a Sphere

作者：Hu, Yingxiang; Xu, Hongwei^*

来源：Journal of Geometric Analysis, 2017, 27(3): 2472-2489.

DOI：10.1007/s12220-017-9769-4

• Compact hypersurfaces
• First eigenvalue
• Differentiable pinching theorem

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,M- g) be a closed, connected, and oriented Riemannian manifold isometrically immersed by phi into Sn+1. Let q > n and A > 0 be some real numbers satisfying |M|(1/n)1 + parallel to B parallel to(q)) <= A. Suppose that phi(M) subset of B(p(0), R) where p(0) is a center of gravity of M and radius R < pi/2. We prove that there exists a positive constant epsilon depending on q, n, R, and A such that if n(1 + parallel to H parallel to(2))(infinity) - epsilon <= lambda(1), then M is diffeomorphic to S-n. Furthermore, phi(M) is starshaped with respect to p(0), almost-isometric to the geodesic sphere S(p(0), R-0) where (sic).

• 出版日期2017-7
• 单位浙江大学