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arXiv · 2405.18233

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

Marcio Batista·Marcos P. Cavalcante·Luiz R. Melo
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Abstract

In this paper, we obtain geometric upper bounds for the first eigenvalue $λ_1(J)$ 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 $λ_1(J)$ and the curvature of the ambient space. We also address the Jacobi--Steklov problem, proving geometric upper bounds for its first eigenvalue $σ_1(J)$ and deriving rigidity results related to the length of the boundary. Additionally, we present some results in higher dimensions related to the Yamabe invariants.

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Marcio Batista, Marcos P. Cavalcante, Luiz R. Melo. 2026-07-16. First Eigenvalue of Jacobi operator and Rigidity Results for Constant Mean Curvature Hypersurfaces. https://arxiv.org/abs/2405.18233

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