Moduli spaces of critical Riemannian metrics with L^{n/2} norm curvature bounds

TL;DR

This paper studies the structure and compactification of moduli spaces of critical Riemannian metrics, particularly extremal Kähler metrics, under specific geometric bounds, revealing their orbifold compactifications.

Contribution

It establishes conditions under which the moduli space of extremal Kähler metrics can be compactified by including orbifolds with finitely many singularities.

Findings

Moduli space can be compactified with orbifolds under volume, curvature, and Sobolev bounds.

Results extend to certain other classes of critical Riemannian metrics.

Provides a framework for understanding limits of extremal Kähler metrics.

Abstract

We consider the moduli space of the extremal K\"ahler metrics on compact manifolds. We show that under the conditions of two-sided total volume bounds, $L^{n\over2}$-norm bounds on $\Riem$, and Sobolev constant bounds, this Moduli space can be compactified by including (reduced) orbifolds with finitely many singularities. Most of our results go through for certain other classes of critical Riemannian metrics.