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On the behaviour of harmonic functions on Riemannian cones

Jean C. Cortissoz

Abstract

We discuss the behavior of harmonic functions on Riemannian cones as defined below and Lioville's theorem.

On the behaviour of harmonic functions on Riemannian cones

Abstract

We discuss the behavior of harmonic functions on Riemannian cones as defined below and Lioville's theorem.
Paper Structure (4 sections, 4 theorems, 33 equations)

This paper contains 4 sections, 4 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. An abstract Liouville's theorem
  4. Applications

Key Result

Theorem 1

Let $\left(M,g\right)$ be a Riemannian manifold with $Ric\left(g\right)\geq 0$. Let $O$ be a fixed point on $M$ and for $x\in M$ define $r_x=d\left(O,x\right)$, the Riemannian distance from $x$ to $O$. Let $u:M\longrightarrow \mathbb{R}$ be a harmonic function. If $u=o\left(r_x\right)$, then $u$ is

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Theorem 3: Anderson83Sullivan83
  • Theorem 4