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Fenchel-Willmore and Sobolev-type inequalities for submanifolds in non-negatively curved manifolds

Meng Ji, Kwok-Kun Kwong

Abstract

In this paper, we uncover a novel connection between the Fenchel-Willmore inequality and a new logarithmic Sobolev inequality for mean-convex submanifolds immersed in non-negatively curved manifolds with Euclidean volume growth. Building on this connection, we establish extensions of the Fenchel-Willmore inequality to submanifolds with boundary and to complete non-compact submanifolds. In addition, we derive a sharp Sobolev-type inequality for submanifolds in the same setting. These Sobolev-type inequalities admit a number of applications, including topological consequences in the surface case.

Fenchel-Willmore and Sobolev-type inequalities for submanifolds in non-negatively curved manifolds

Abstract

In this paper, we uncover a novel connection between the Fenchel-Willmore inequality and a new logarithmic Sobolev inequality for mean-convex submanifolds immersed in non-negatively curved manifolds with Euclidean volume growth. Building on this connection, we establish extensions of the Fenchel-Willmore inequality to submanifolds with boundary and to complete non-compact submanifolds. In addition, we derive a sharp Sobolev-type inequality for submanifolds in the same setting. These Sobolev-type inequalities admit a number of applications, including topological consequences in the surface case.

Paper Structure

This paper contains 7 sections, 21 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Proofs of non-compact Fenchel-Willmore inequalities
  3. Proof of Theorem \ref{['thm log sob ineq']}
  4. Proof of Theorem \ref{['thm log sob intro equality']}
  5. Proof of Theorem \ref{['thm nonneg Sobolev']}
  6. Some applications of Theorem \ref{['thm nonneg Sobolev']}
  7. Proof of Theorem \ref{['thm nonneg Sobolev equality']}

Key Result

Theorem 1.1

Let $n, m \in \mathbb{N}$, and $(M, g)$ be a complete non-compact Riemannian manifold of dimension $n+m$ with nonnegative sectional curvature and positive asymptotic volume ratio $\theta$. Suppose that $\Sigma$ is a compact $n$-dimensional submanifold immersed in $M$ (possibly with boundary $\partia where If $m\le 3$, the equality holds if and only if $\Sigma$ is connected, umbilical and with no

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • proof : Proof of Theorem \ref{['thm non-compact fenchel willmore']}
  • Remark 2.1
  • Definition 2.2
  • ...and 28 more