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

In this paper, we obtain geometric upper bounds for the first eigenvalue $λ_1(J)$ of the Jacobi operator for both closed and compact with boundary hypersurfaces 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.

💡 Research Summary

The paper investigates the first eigenvalues of the Jacobi operator and the Jacobi–Steklov problem on constant mean curvature (CMC) hypersurfaces, both closed and with free boundary, and derives sharp geometric upper bounds that lead to rigidity results.

The authors begin by recalling the Jacobi operator (J=\Delta+\operatorname{Ric}_M(N,N)+|A|^2) associated with a two‑sided CMC hypersurface (\Sigma^n\subset M^{n+1}). For a function (u\in H^1(\Sigma)) the quadratic form
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