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Stability of vertical and radial graphs in the Euclidean space with density

Rafael López

Abstract

It is proved that vertical graphs and radial graphs are strongly stable for a certain type of densities in Euclidean space ${\mathbb R}^{n+1}$. Particular cases of these densities include translators, expanders and singular minimal hypersurfaces. Using techniques of calibrations, it is also proved that for densities depending on a spatial coordinate, stationary vertical graphs are weighted minimizers in a certain class of hypersurfaces.

Stability of vertical and radial graphs in the Euclidean space with density

Abstract

It is proved that vertical graphs and radial graphs are strongly stable for a certain type of densities in Euclidean space . Particular cases of these densities include translators, expanders and singular minimal hypersurfaces. Using techniques of calibrations, it is also proved that for densities depending on a spatial coordinate, stationary vertical graphs are weighted minimizers in a certain class of hypersurfaces.
Paper Structure (3 sections, 16 theorems, 52 equations)

This paper contains 3 sections, 16 theorems, 52 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Stability of vertical graphs and radial graphs
  3. Minimizers of vertical graphs in manifolds with some densities

Key Result

Lemma 2.1

Let $\mathsf{a}\in\mathbb{R}^{n+1}$, $\lvert \mathsf{a}\rvert=1$. If $\Sigma$ is a $\phi$-stationary surface, then the function $g=\langle N,\mathsf{a}\rangle$ satisfies where $\mathsf{D}$ is the Levi-Civita connection in $\mathbb{R}^{n+1}$ and $\{e_i:1\leq i\leq n\}$ is a local orthonormal frame on $\Sigma$.

Theorems & Definitions (27)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Corollary 2.4
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • proof
  • ...and 17 more