Harmonic nets in metric spaces

TL;DR

This paper introduces harmonic nets, a new concept for harmonic maps from weighted graphs into metric spaces with unique centers of gravity, and proves their existence through an iterative geometric process.

Contribution

It develops the theory of harmonic nets in metric spaces with unique centers of gravity and provides an existence proof via an iterative construction.

Findings

Harmonic nets exist in certain metric spaces.

An iterative geometric process converges to harmonic maps.

Application to spaces with upper curvature bounds.

Abstract

We investigate harmonic maps from weighted graphs into metric spaces that locally admit unique centers of gravity, like Alexandrov spaces with upper curvature bounds. We prove an existence result by constructing an iterative geometric process that converges to such maps, called harmonic nets for short.