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Warped area minimal hypersurface and warped product metric

Yukai Sun

Abstract

By studying the warped(or weighted) area-minimizing hypersurface, we prove that the metric can be locally split as a warped product metric under the spectral Ricci or spectral scalar curvature lower bound condition.

Warped area minimal hypersurface and warped product metric

Abstract

By studying the warped(or weighted) area-minimizing hypersurface, we prove that the metric can be locally split as a warped product metric under the spectral Ricci or spectral scalar curvature lower bound condition.
Paper Structure (4 sections, 13 theorems, 62 equations)

This paper contains 4 sections, 13 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Basic fact about warped or weighted minimal hypersurface
  3. Spectral Ricci case
  4. Spectral scalar curvature case

Key Result

Theorem 1.2

Let $(M^{n},g)$ be a smooth compact connected Riemannian manifold satisfying where $u$ is a smooth positive function on $M$ and $3\leq n\leq 7$. Suppose further that $H_{n-1}(M,\mathbb{Z})\neq 0$. Then there exists a non-trivial homology class $[\Sigma]\in H_{n-1}(M,\mathbb{Z})$(i.e., $[\Sigma]\neq 0$) such that:

Theorems & Definitions (27)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • proof : Proof of Theorem \ref{['Thm-ricci']}
  • ...and 17 more