Regularity and convergence of 4-dimensional extremal Kahler metrics

TL;DR

This paper proves regularity and compactness results for 4-dimensional extremal Kähler metrics, showing that under certain curvature bounds, sequences of such metrics converge to orbifold limits with controlled singularities.

Contribution

It establishes a new regularity theorem and a weak compactness result for 4-dimensional extremal Kähler metrics, linking curvature bounds to Gromov-Hausdorff convergence.

Findings

Sectional curvature bounds under small $L^2$ curvature conditions.

Sequences with bounded Calabi energy and scalar curvature have convergent subsequences.

Limits are Riemannian orbifolds with finitely many singularities.

Abstract

We establish a regularity result for the metric on any 4-dimensional extremal K\"ahler manifold, and a weak compactness theorem on the space of such metrics. Specifically, the sectional curvature at a point is bounded when the quantity $L^2(|\Riem|)$ in a surrounding ball is sufficiently small compared to the pointwise norm of its scalar curvature. Consequently sequences of 4-dimensional extremal K\"ahler metrics with uniformly bounded Calabi energies and scalar curvature have convergent subsequences in the Gromov-Hausdorff topology. Gromov-Hausdorff limits are length spaces with the structure of Riemannian orbifolds away from finitely many point-like singularities of unknown structure.