A Framework for Gluing Harmonic Maps

TL;DR

This paper develops a mathematical framework for constructing extended harmonic maps between Riemann surfaces and Riemannian manifolds using a gluing technique, advancing the understanding of harmonic map theory.

Contribution

It introduces a novel gluing construction method for extended harmonic maps, providing new tools for analyzing harmonic maps between complex geometric structures.

Findings

Established a new gluing map construction for harmonic maps

Provided conditions under which the gluing construction is valid

Enhanced the theoretical understanding of harmonic map extensions

Abstract

In this paper, we study the gluing construction of the extended harmonic maps between Riemannian manifolds. Harmonic maps are critical points of the energy functional. We construct the gluing map of the extended harmonic maps from Riemann surfaces to some Riemannian manifold $N$ under certain conditions.