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Bernstein and half-space properties for minimal graphs under Ricci lower bounds

Giulio Colombo, Marco Magliaro, Luciano Mari, Marco Rigoli

Abstract

In this paper, we prove a new gradient estimate for minimal graphs defined on domains of a complete manifold with Ricci curvature bounded from below. In particular, we show that positive, entire minimal graphs on manifolds with non-negative Ricci curvature are constant, and that complete, parabolic manifolds with Ricci curvature bounded from below have the half-space property. We avoid the need of sectional curvature bounds on $M$ by exploiting a form of the Ahlfors-Khas'minskii duality in nonlinear potential theory.

Bernstein and half-space properties for minimal graphs under Ricci lower bounds

Abstract

In this paper, we prove a new gradient estimate for minimal graphs defined on domains of a complete manifold with Ricci curvature bounded from below. In particular, we show that positive, entire minimal graphs on manifolds with non-negative Ricci curvature are constant, and that complete, parabolic manifolds with Ricci curvature bounded from below have the half-space property. We avoid the need of sectional curvature bounds on by exploiting a form of the Ahlfors-Khas'minskii duality in nonlinear potential theory.

Paper Structure

This paper contains 5 sections, 7 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['teo_graph_soloricci_intro']}
  4. Remarks: harmonic functions and minimal graphs
  5. Bernstein and half-space properties: proof of Theorem \ref{['mainteo_tiporss']}

Key Result

Theorem 1

Let $M$ be a complete manifold of dimension $m \ge 2$.

Theorems & Definitions (23)

  • Definition 1
  • Theorem 1: rosenbergschulzespruck
  • Remark 1
  • Theorem 2
  • Remark 2: Manifolds with slow volume growth
  • Theorem 3
  • Remark 3
  • Remark 4
  • Theorem 4
  • Remark 5
  • ...and 13 more