Recognizing shape via 1st eigenvalue, mean curvature and upper curvature\n bound

2019/05/05 by Yingxiang Hu, Hu, Yingxiang, Shicheng Xu +1
Mathematics · #53C20 #53C21 #53C24 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.1905.01664

openalex publication_date 2019/05/05 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28

Abstract

Let Mⁿ be a closed immersed hypersurface lying in a contractible ball\nB(p,R) of the ambient (n+1)-manifold Nⁿ⁺¹. We prove that, by pinching\nHeintze-Reilly's inequality via sectional curvature upper bound of B(p,R),\n1st eigenvalue and mean curvature of M, not only M is Hausdorff close to a\ngeodesic sphere S(p₀,R₀) in N, but also the ``enclosed'' ball\nB(p₀,R₀) is close to be of constant curvature, provided with a uniform\ncontrol on the volume and mean curvature of M. We raise a conjecture for M\nto be a diffeomorphic sphere, and give some positive partial answer.\n

Citations

• On the proof of the positive mass conjecture in general relativity
• Lower Bounds on Ricci Curvature and the Almost Rigidity of Warped Products
• Collapsing of Riemannian manifolds and eigenvalues of Laplace operator

Related

• 1st eigenvalue pinching for convex hypersurfaces in a Riemannian manifold
• Total squared mean curvature of immersed submanifolds in a negatively curved space
• Embedded constant m^th mean curvature hypersurfaces on spheres
• A note on the first eigenvalue of spherically symmetric manifolds
• A new characterization of geodesic spheres in the Hyperbolic space