The paper aims to prove that a locally mass-maximizing free boundary minimal disk in a negatively curved three-manifold must sit in a half anti-de Sitter Schwarzschild model, but the proof's sign analysis is reversed.
First Eigenvalue of Jacobi operator and Rigidity Results for Constant Mean Curvature Hypersurfaces
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abstract
In this paper, we obtain geometric upper bounds for the first eigenvalue $\lambda_1(J)$ 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 $\lambda_1(J)$ and the curvature of the ambient space. We also address the Jacobi--Steklov problem, proving geometric upper bounds for its first eigenvalue $\sigma_1(J)$ 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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math.DG 1years
2025 1verdicts
REJECT 1representative citing papers
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Modified Hawking mass and rigidity of three-manifolds with boundary
The paper aims to prove that a locally mass-maximizing free boundary minimal disk in a negatively curved three-manifold must sit in a half anti-de Sitter Schwarzschild model, but the proof's sign analysis is reversed.