Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Rigidity of surfaces with nonpositive Euler characteristic by the second eigenvalue of the Jacobi operator

Márcio Batista, Marcos P. Cavalcante, Abraão Mendes, Ivaldo Nunes

TL;DR

The paper establishes sharp bounds for the second eigenvalue of the Jacobi operator on closed surfaces with nonpositive Euler characteristic embedded in spheres, linking spectral data to Willmore energy and rigidity. It extends these results to product manifolds $\mathbb{S}^1(r)\times\mathbb{S}^2(s)$, showing that for $r\ge s$ the second Jacobi eigenvalue is nonpositive for positive-genus surfaces and equality characterizes the totally geodesic torus when $r=s$. The methodology combines conformal area techniques, extrinsic-geometry analysis, and Willmore-energy inequalities to derive both upper bounds and sharp rigidity statements, with explicit flat-torus examples demonstrating sharpness. Overall, the work provides a unified spectral-geometric framework for characterizing extremal surfaces in high-dimensional spheres and product spaces, with potential implications for stability and rigidity in geometric analysis.

Abstract

In this paper, we investigate the spectral properties of the Jacobi operator for immersed surfaces with nonpositive Euler characteristic, extending previous results in the field. We first prove a sharp upper bound for the second eigenvalue of the Jacobi operator for compact surfaces with nonpositive Euler characteristic that are fully immersed in the Euclidean sphere, and then we classify all such surfaces attaining this upper bound. Furthermore, we demonstrate that totally geodesic tori maximize the second eigenvalue among all compact orientable surfaces with positive genus in the product space $\mathbb{S}^1(r) \times \mathbb{S}^2(s)$.

Rigidity of surfaces with nonpositive Euler characteristic by the second eigenvalue of the Jacobi operator

TL;DR

The paper establishes sharp bounds for the second eigenvalue of the Jacobi operator on closed surfaces with nonpositive Euler characteristic embedded in spheres, linking spectral data to Willmore energy and rigidity. It extends these results to product manifolds , showing that for the second Jacobi eigenvalue is nonpositive for positive-genus surfaces and equality characterizes the totally geodesic torus when . The methodology combines conformal area techniques, extrinsic-geometry analysis, and Willmore-energy inequalities to derive both upper bounds and sharp rigidity statements, with explicit flat-torus examples demonstrating sharpness. Overall, the work provides a unified spectral-geometric framework for characterizing extremal surfaces in high-dimensional spheres and product spaces, with potential implications for stability and rigidity in geometric analysis.

Abstract

In this paper, we investigate the spectral properties of the Jacobi operator for immersed surfaces with nonpositive Euler characteristic, extending previous results in the field. We first prove a sharp upper bound for the second eigenvalue of the Jacobi operator for compact surfaces with nonpositive Euler characteristic that are fully immersed in the Euclidean sphere, and then we classify all such surfaces attaining this upper bound. Furthermore, we demonstrate that totally geodesic tori maximize the second eigenvalue among all compact orientable surfaces with positive genus in the product space .

Paper Structure

This paper contains 12 sections, 9 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Area estimates for surfaces in the unit sphere
  4. Extrinsic geometry of $\mathbb{S}^1(r) \times \mathbb{S}^n(\sqrt{1 - r^2})$ in $\mathbb{S}^{n+2}$
  5. Spectral characterization of surfaces with nonpositive Euler characteristic in higher-dimensional spheres
  6. Minimal tori in spheres
  7. The Klein bottle as a minimal surface in the sphere
  8. Eigenvalue estimate via Willmore energy
  9. Spectral characterization of surfaces in $\mathbb{S}^1(r)\times\mathbb{S}^2(s)$
  10. Examples of flat tori in $\mathbb{S}^1(r) \times \mathbb{S}^2(s)$, with $r \leq s$
  11. Case 1: $r \geq t$
  12. Case 2: $r \leq t$

Key Result

Theorem A

Let $\Sigma$ be a closed surface with nonpositive Euler characteristic, fully immersed in $\mathbb{S}^n$. Then, the second eigenvalue of the Jacobi operator of $\Sigma$ in $\mathbb{S}^n$, satisfies Furthermore, if $\lambda_2(L) = -2$, then $\Sigma$ is orientable and one of the following holds:

Theorems & Definitions (15)

  • Theorem A
  • Theorem B
  • Remark 1
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 3.1: El Soufi-Ilias, ElSoufiIlias2
  • Theorem 3.2: El Soufi-Giacomini-Jazar, EGJ06
  • Remark 2
  • ...and 5 more