Compactness of harmonic maps of surfaces with regular nodes

TL;DR

This paper establishes a compactness theorem for harmonic maps from surfaces with regular nodes, using moduli space techniques, and proves conditions under which energy concentration and bubbling phenomena are controlled.

Contribution

It introduces a general compactness theorem for harmonic maps on complex curves with nodes, extending previous results to more general degenerations.

Findings

Neck regions have zero energy and length under certain conditions.

Energy identity and zero distance bubbling hold for maps from spheres.

Provides a framework for analyzing harmonic maps on degenerating surfaces.

Abstract

In this paper, we formulate and prove a general compactness theorem for harmonic maps using Deligne-Mumford moduli space and families of curves. The main theorem shows that given a sequence of harmonic maps over a sequence of complex curves, there is a family of curves and a subsequence such that both the domains and the maps converge off the set of "non-regular" nodes. This provides a sufficient condition for a neck having zero energy and zero length. As a corollary, the following known fact can be proved: If all domains are diffeomorphic to $S^2$, both energy identity and zero distance bubbling hold.