Riemannian Polyhedra and Liouville-type Theorems for Harmonic maps

TL;DR

This paper investigates harmonic maps from Riemannian polyhedra to non-positively curved spaces, establishing Liouville-type theorems and vanishing results under various geometric conditions.

Contribution

It extends Liouville-type theorems to Riemannian polyhedra and non-smooth settings, providing new results for harmonic maps in Alexandrov spaces.

Findings

Liouville-type theorems for harmonic maps from pseudomanifolds with non-negative Ricci curvature.

Vanishing results for harmonic maps on 2-parabolic Riemannian polyhedra.

Extension of classical results to non-smooth geometric contexts.

Abstract

This paper is a study of harmonic maps from Riemannian polyhedra to (locally) non-positively curved geodesic spaces in the sense of Alexandrov. We prove Liouville-type theorems for subharmonic functions and harmonic maps under two different assumptions on the source space. First we prove the analogue of the Schoen-Yau Theorem on a complete (smooth) pseudomanifolds with non-negative Ricci curvature. Then we study 2-parabolic admissible Riemannian polyhedra and prove some vanishing results on them.