First Eigenvalue of Jacobi operator and Rigidity Results for Constant Mean Curvature Hypersurfaces

2024/05/28 by Marcio Batista, Marcos P. Cavalcante, Batista, Marcio +3
#math.DG #math.AP #math.SP

paper · pdf · doi:10.48550/arxiv.2405.18233

Abstract

In this paper, we obtain geometric upper bounds for the first eigenvalue λ₁(J) of the Jacobi operator for both closed hypersurfaces and compact hypersurfaces with boundary having constant mean curvature (CMC). As an application, we derive new rigidity results for the area of CMC hypersurfaces under suitable conditions on λ₁(J) and the curvature of the ambient space. We also address the Jacobi--Steklov problem, proving geometric upper bounds for its first eigenvalue σ₁(J) and deriving rigidity results related to the length of the boundary. Additionally, we present some results in higher dimensions related to the Yamabe invariants.

Citations

• First stability eigenvalue of singular hypersurfaces with constant mean curvature in the unit sphere
• Index of the critical catenoid
• First eigenvalue of a Jacobi operator of hypersurfaces with a constant scalar curvature
• A spectral characterization of the H(r)-torus by the first stability eigenvalue
• First stability eigenvalue characterization of Clifford hypersurfaces
• Existence of Incompressible Minimal Surfaces and the Topology of Three Dimensional Manifolds with Non-Negative Scalar Curvature
• The unknottedness of minimal embeddings
• Rigidity of compact manifolds with boundary and nonnegative Ricci curvature

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• Rigidity of surfaces with nonpositive Euler characteristic by the second eigenvalue of the Jacobi operator
• On the first stability eigenvalue of surfaces with constant weighted mean curvature