Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Gauss map of hypersurfaces with constant weighted mean curvature in the Gaussian space

Michael Gomez, Matheus Vieira

Abstract

In this paper we study the Gauss map of hypersurfaces with constant weighted mean curvature in the Gaussian space. We show that if the image of the Gauss map is in a closed hemisphere, then the hypersurface is a hyperplane or a generalized cylinder. We also show that if the image of the Gauss map is in $S^{n}\setminus\bar{S}_{+}^{n-1}$, then the hypersurface is a hyperplane. This generalizes previous results for self-shrinkers obtained by Ding-Xin-Yang.

The Gauss map of hypersurfaces with constant weighted mean curvature in the Gaussian space

Abstract

In this paper we study the Gauss map of hypersurfaces with constant weighted mean curvature in the Gaussian space. We show that if the image of the Gauss map is in a closed hemisphere, then the hypersurface is a hyperplane or a generalized cylinder. We also show that if the image of the Gauss map is in , then the hypersurface is a hyperplane. This generalizes previous results for self-shrinkers obtained by Ding-Xin-Yang.
Paper Structure (8 sections, 8 theorems, 45 equations)

This paper contains 8 sections, 8 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Smooth metric measure spaces
  4. Weighted mean curvature
  5. Hypersurfaces in the Gaussian space
  6. Lemmas
  7. Proof of Theorem \ref{['thm:hemisphere']}
  8. Proof of Theorem \ref{['thm:semicircle']}

Key Result

Theorem 1

Let $M$ be a complete hypersurface with constant weighted mean curvature $\lambda$ properly immersed in the Gaussian space $R^{n+1}$. Suppose that the image of the Gauss map is in the closed hemisphere $x_{n+1}\geq0$. Then either: (i) $M$ is a hyperplane, or (ii) $M$ splits isometrically as $M^{n}=\

Theorems & Definitions (8)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Theorem 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8