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The rigidity of minimal Legendrian submanifolds in the Euclidean spheres via eigenvalues of fundamental matrices

Pei-Yi Wu, Ling Yang

Abstract

In this paper, we study the rigidity problem for compact minimal Legendrian submanifolds in the unit Euclidean spheres via eigenvalues of fundamental matrices, which measure the squared norms of the second fundamental form on all normal directions. By using Lu's inequality on the upper bound of the squared norm of Lie brackets of symmetric matrices, we establish an optimal pinching theorem for such submanifolds of all dimensions, giving a new characterization for the Calabi tori. This pinching condition can also be described by the eigenvalues of the Ricci curvature tensor. Moreover, when the third large eigenvalue of the fundamental matrix vanishes everywhere, we get an optimal rigidity theorem under a weaker pinching condition.

The rigidity of minimal Legendrian submanifolds in the Euclidean spheres via eigenvalues of fundamental matrices

Abstract

In this paper, we study the rigidity problem for compact minimal Legendrian submanifolds in the unit Euclidean spheres via eigenvalues of fundamental matrices, which measure the squared norms of the second fundamental form on all normal directions. By using Lu's inequality on the upper bound of the squared norm of Lie brackets of symmetric matrices, we establish an optimal pinching theorem for such submanifolds of all dimensions, giving a new characterization for the Calabi tori. This pinching condition can also be described by the eigenvalues of the Ricci curvature tensor. Moreover, when the third large eigenvalue of the fundamental matrix vanishes everywhere, we get an optimal rigidity theorem under a weaker pinching condition.
Paper Structure (11 sections, 13 theorems, 132 equations)

This paper contains 11 sections, 13 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Legendrian submanifolds in the unit spheres
  4. The Simons type identity
  5. On Lu's inequality
  6. An optimal pinching theorem on $|B|^2+\lambda_2$
  7. A Simons type integral inequality
  8. A characterization of the Calabi tori
  9. Compact minimal Legendrian submanifolds with $\lambda_3\equiv 0$
  10. The rank of Gauss maps
  11. A structure theorem for $3$-dimensional minimal Legendrian submanifolds

Key Result

Theorem 1.1

Let $M$ be an $n$-dimensional compact minimal submanifold in $S^{n+m}$, denote by $|B|^2$ be the square norm of the second fundamental form $B$ of $M$, then As a corollary, the pinching condition $0\leq |B|^2\leq \frac{n}{2-\frac{1}{m}}$ forces $|B|^2\equiv 0$ or $|B|^2\equiv \frac{n}{2-\frac{1}{m}}$.

Theorems & Definitions (16)

  • Theorem 1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Remark 2.1
  • ...and 6 more