An Observation on the Dirichlet problem at infinity in Riemannian cones

TL;DR

This paper establishes a new sufficient condition for solving the Dirichlet problem at infinity in Riemannian cones, extending classical criteria through elementary methods and connecting to Milnor's classification of parabolic surfaces.

Contribution

It introduces a novel sufficient condition for solvability in Riemannian cones, generalizing classical results for manifolds with special metrics.

Findings

New sufficient condition for Dirichlet problem solvability

Generalizations of classical criteria for specific Riemannian manifolds

Elementary proof using separation of variables and ODE comparison

Abstract

In this short paper we show a sufficient condition for the solvability of the Dirichlet problem at infinity in Riemannian cones (as defined below).This condition is related to a celebrated result of Milnor that classifies parabolic surfaces. When applied tosmooth Riemannian manifolds with a special type of metrics (which generalise rotational symmetry) we obtain generalisations of classical criteria for the solvability of the Dirichlet problem at infinity. Our proof is short and elementary: it uses separation of variables and comparison arguments for ODE's.