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A Moser-Bernstein problem for Riemannian warped products

Alma L. Albujer, Jónatan Herrera, Rafael M. Rubio

Abstract

In this work we deal with an elliptic non-linear problem, which arises naturally from Riemannian geometry. This problem has clasically been studied in the the Euclidean $n$-dimensional space and it is known as the Moser-Bernstein problem. Nevertheless we solve this type of problems in a wide family of Riemannian manifolds, constructed as Riemannian warped products. More precicely, we study the entire solutions to the minimal hypersurface equation in a Riemannian warped product $M=P\times_h\mathbb{R}$, where $P$ is a complete Riemannian parabolic manifold and $h$ a positive smooth function on $P$.

A Moser-Bernstein problem for Riemannian warped products

Abstract

In this work we deal with an elliptic non-linear problem, which arises naturally from Riemannian geometry. This problem has clasically been studied in the the Euclidean -dimensional space and it is known as the Moser-Bernstein problem. Nevertheless we solve this type of problems in a wide family of Riemannian manifolds, constructed as Riemannian warped products. More precicely, we study the entire solutions to the minimal hypersurface equation in a Riemannian warped product , where is a complete Riemannian parabolic manifold and a positive smooth function on .
Paper Structure (8 sections, 7 theorems, 27 equations)

This paper contains 8 sections, 7 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Setting up
  4. Parabolic Riemannian manifolds
  5. The mean curvature of the graph
  6. Main Results
  7. New Moser-Bernstein theorems
  8. Acknowledgements

Key Result

Proposition 3.1

(Grigor_yan_1999) If two manifolds $(P,\sigma)$ and $(P',\sigma')$ are quasi-isometric, then both are parabolic or not simultaneously.

Theorems & Definitions (10)

  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Theorem 4.1
  • proof
  • Corollary 4.1
  • Example 4.2
  • Proposition 4.3
  • Theorem 5.1
  • Theorem 5.2