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

Marcio Batista, Marcos P. Cavalcante, Luiz R. Melo

TL;DR

This work derives geometric upper bounds for the first Jacobi eigenvalue $λ_1(J)$ and the Jacobi–Steklov eigenvalue $σ_1(J)$ for closed and free-boundary CMC hypersurfaces, respectively, and translates these bounds into sharp area-rigidity results. By exploiting the Gauss equation and Rayleigh-type variational formulas, the authors obtain rigidity statements that identify hemispherical models (and, in higher dimensions, ball-models) under equality, under natural ambient curvature assumptions. The Jacobi–Steklov theory is developed via a Dirichlet-to-Neumann map under a Dirichlet-positivity condition, yielding a geometric bound for $σ_1(J)$ and equality conditions tied to total umbilicity and boundary geometry. Finally, the paper connects these spectral bounds to Escobar’s Yamabe invariants, establishing relations between eigenvalues and Yamabe constants that yield additional rigidity conclusions in both lower and higher dimensions. Together, these results reinforce the deep link between spectral data of the Jacobi operator, area rigidity for CMC hypersurfaces, and global geometric constraints from ambient curvature and Yamabe-type invariants.

Abstract

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.

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

TL;DR

This work derives geometric upper bounds for the first Jacobi eigenvalue and the Jacobi–Steklov eigenvalue for closed and free-boundary CMC hypersurfaces, respectively, and translates these bounds into sharp area-rigidity results. By exploiting the Gauss equation and Rayleigh-type variational formulas, the authors obtain rigidity statements that identify hemispherical models (and, in higher dimensions, ball-models) under equality, under natural ambient curvature assumptions. The Jacobi–Steklov theory is developed via a Dirichlet-to-Neumann map under a Dirichlet-positivity condition, yielding a geometric bound for and equality conditions tied to total umbilicity and boundary geometry. Finally, the paper connects these spectral bounds to Escobar’s Yamabe invariants, establishing relations between eigenvalues and Yamabe constants that yield additional rigidity conclusions in both lower and higher dimensions. Together, these results reinforce the deep link between spectral data of the Jacobi operator, area rigidity for CMC hypersurfaces, and global geometric constraints from ambient curvature and Yamabe-type invariants.

Abstract

In this paper, we obtain geometric upper bounds for the first eigenvalue 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 and the curvature of the ambient space. We also address the Jacobi--Steklov problem, proving geometric upper bounds for its first eigenvalue and deriving rigidity results related to the length of the boundary. Additionally, we present some results in higher dimensions related to the Yamabe invariants.
Paper Structure (12 sections, 15 theorems, 65 equations)

This paper contains 12 sections, 15 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. The First Eigenvalue of the Jacobi Operator
  3. Rigidity results for the area of CMC surfaces
  4. Proof of Theorem \ref{['T1']}
  5. Proof of Theorem \ref{['T2']}
  6. Proof of Theorem \ref{['T3']}
  7. Proof of Theorem \ref{['T4']}
  8. The Jacobi--Steklov problem
  9. The boundary value problem and the Dirichlet-to-Neumann operator
  10. Variational characterizations
  11. A geometric upper bound for $\sigma_1(J)$
  12. Estimates involving the Yamabe invariants

Key Result

Theorem 1.1

Let $M^3$ be a Riemannian manifold with Ricci curvature $\operatorname{Ric}_M \geq 2$ and convex boundary in the sense that its second fundamental form satisfies $\mathrm I\!\mathrm I^{\partial M} \geq 0$. Let $\Sigma^2 \subset M^3$ be an immersed and orientable surface with constant mean curvature

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • ...and 18 more