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Small spheres with prescribed nonconstant mean curvature in Riemannian manifolds

Alberto Enciso, Antonio J. Fernández, Daniel Peralta-Salas

Abstract

Given a function $f$ on a smooth Riemannian manifold without boundary, we prove that if $p \in M$ is a non-degenerate critical point of $f$, then a neighborhood of $p$ contains a foliation by spheres with mean curvature proportional to $f$. This foliation is essentially unique. The nondegeneracy assumption can be substantially relaxed, at the expense of losing the property that the family of spheres with prescribed mean curvature defines a foliation.

Small spheres with prescribed nonconstant mean curvature in Riemannian manifolds

Abstract

Given a function on a smooth Riemannian manifold without boundary, we prove that if is a non-degenerate critical point of , then a neighborhood of contains a foliation by spheres with mean curvature proportional to . This foliation is essentially unique. The nondegeneracy assumption can be substantially relaxed, at the expense of losing the property that the family of spheres with prescribed mean curvature defines a foliation.
Paper Structure (4 sections, 11 theorems, 58 equations)

This paper contains 4 sections, 11 theorems, 58 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. The non-degenerate case: proof of Theorem \ref{['mainintro']}
  4. The degenerate case: proof of Theorem \ref{['mainintro-index']}

Key Result

Theorem 1.1

If $p \in M$ is a non-degenerate critical point of the scalar curvature of $M$, then a punctured neighborhood of $p$ contains a smooth foliation $\mathscr{F}:= \{S_r : 0 < r < \delta\}$ by CMC spheres. The mean curvature of each $S_r$ is $H = \overline{c}/r$ for some constant $\overline{c}> 0$.

Theorems & Definitions (18)

  • Theorem 1.1: Ye, 1991
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2: Fo2001
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 8 more