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A conformal invariant and its application to the nonexistence of minimal submanifolds

Hang Chen

Abstract

Let $(M^m,g)$ be an $m$-dimensional closed Riemannian manifold with non-negative sectional curvatures, $m\ge 3$. We define a conformal invariant and prove that, if the conformal invariant is bounded from above by a constant depending only on $m$, then there are no closed $n$-dimensional stable minimal submanifolds in $M$ for all $ξ(m)\le n\le m-2$, where $ξ(m)=1$ when $3\le m\le 5$ and $ξ(m)=2$ when $m\ge 6$. In particular, a conformal $m$-sphere with non-negative sectional curvatures does not admit any closed $n$-dimensional stable minimal submanifold for all $ξ(m)\le n\le m-2$.

A conformal invariant and its application to the nonexistence of minimal submanifolds

Abstract

Let be an -dimensional closed Riemannian manifold with non-negative sectional curvatures, . We define a conformal invariant and prove that, if the conformal invariant is bounded from above by a constant depending only on , then there are no closed -dimensional stable minimal submanifolds in for all , where when and when . In particular, a conformal -sphere with non-negative sectional curvatures does not admit any closed -dimensional stable minimal submanifold for all .
Paper Structure (7 sections, 12 theorems, 78 equations)

This paper contains 7 sections, 12 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Conformal relations
  5. Some calculations
  6. Proof of main theorems
  7. The conformal second fundamental form of ellipsoids

Key Result

Theorem 1.1

Let $\Sigma^n$ be a closed minimal submanifold immersed in $M^m$. Then $\Sigma$ is stable if and only if $\Sigma$ is one of the cases listed in the following table.

Theorems & Definitions (24)

  • Theorem 1.1
  • Conjecture 1.2: LS73
  • Theorem 1.3: FZ23
  • Definition 1.4
  • Remark 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Remark 1.10
  • ...and 14 more