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Pinching rigidity of minimal surfaces in spheres

Weiran Ding, Jianquan Ge, Fagui Li

Abstract

In this paper we give a pinching theorem of the Simon conjecture in the case s=3 and also give a new proof of the cases s=1 and s=2 by some Simons-type integral inequalities.

Pinching rigidity of minimal surfaces in spheres

Abstract

In this paper we give a pinching theorem of the Simon conjecture in the case s=3 and also give a new proof of the cases s=1 and s=2 by some Simons-type integral inequalities.

Paper Structure

This paper contains 10 sections, 15 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Notations and Local formulas
  3. The Simon conjecture in the cases $s=1,2$
  4. The first Simons-type identity
  5. The second Simons-type identity
  6. The case $s=3$
  7. The lower bound of $\mathcal{B}_3$
  8. Laplacian of $\mathcal{B}_2$
  9. The third Simons-type identity
  10. Proof of Theorem \ref{['maintheorem']}

Key Result

Theorem A

Let $M$ be a closed surface minimally immersed into $\mathbb{S}^N(1)$.

Theorems & Definitions (34)

  • Conjecture 1: Intrinsic version
  • Conjecture 2: Extrinsic version
  • Theorem A
  • Remark 1
  • Remark 2
  • Corollary B
  • Remark 3
  • Theorem C
  • Theorem 1
  • proof
  • ...and 24 more