Curvature and bubble convergence of harmonic maps

TL;DR

This paper investigates how curvature influences bubble convergence in harmonic maps, establishing universal curvature estimates and demonstrating no curvature loss in neck regions, under the assumption of a Kähler target manifold.

Contribution

It provides a new characterization of bubble formation via curvature excess and offers universal curvature estimates for harmonic maps into Kähler manifolds.

Findings

Curvature excess characterizes bubble formation.

Universal curvature concentration estimates are established.

No curvature loss occurs in neck regions.

Abstract

We explore geometric aspects of bubble convergence for harmonic maps. More precisely, we show that the formation of bubbles is characterised by the local excess of curvature on the target manifold. We give a universal estimate for curvature concentration masses at each bubble point and show that there is no curvature loss in the necks. Our principal hypothesis is that the target manifold is Kaehler.