The compactness result for K\"ahler Ricci solitons

TL;DR

This paper establishes a compactness theorem for sequences of compact K"ahler Ricci gradient shrinking solitons under certain curvature bounds, showing convergence to a K"ahler Ricci soliton orbifold with isolated singularities.

Contribution

It proves a new compactness result for K"ahler Ricci solitons with bounded curvature norms and Perelman's functional, including convergence to orbifolds with singularities.

Findings

Sequences of K"ahler Ricci solitons with bounded curvature norms converge to orbifolds.

Limit spaces are K"ahler Ricci solitons with finitely many isolated singularities.

The convergence preserves the K"ahler Ricci soliton structure away from singularities.

Abstract

In this paper we prove the compactness result for compact K\"ahler Ricci gradient shrinking solitons. If $(M_i,g_i)$ is a sequence of K\"ahler Ricci solitons of real dimension $n \ge 4$, whose curvatures have uniformly bounded $L^{n/2}$ norms, whose Ricci curvatures are uniformly bounded from below and $\mu(g_i,1/2) \ge A$ (where $\mu$ is Perelman's functional), there is a subsequence $(M_i,g_i)$ converging to a compact orbifold $(M_{\infty},g_{\infty})$ with finitely many isolated singularitites, where $g_{\infty}$ is a K\"ahler Ricci soliton metric in an orbifold sense (satisfies a soliton equation away from singular points and smoothly extends in some gauge to a metric satisfying K\"ahler Ricci soliton equation in a lifting around singular points).