Gradient estimates and blow-up analysis for stationary harmonic maps

TL;DR

This paper establishes conditions for gradient estimates and analyzes the structure of singular sets in stationary harmonic maps between Riemannian manifolds, linking energy concentration to geometric regularity.

Contribution

It provides a necessary and sufficient condition for gradient bounds based on energy and characterizes the singular sets when the target manifold lacks harmonic S^2.

Findings

Gradient estimates depend on total energy of maps.

Singular sets are rectifiable if target has no harmonic S^2.

Analysis of defect measures explains energy concentration.

Abstract

For stationary harmonic maps between Riemannian manifolds, we provide a necessary and sufficient condition for the uniform interior and boundary gradient estimates in terms of the total energy of maps. We also show that if analytic target manifolds do not carry any harmonic S^2, then the singular sets of stationary maps are m \leq n - 4 rectifiable. Both of these results follow from a general analysis on the defect measures and energy concentration sets associated with a weakly converging sequence of stationary harmonic maps.