Convergence of compact Ricci solitons

TL;DR

This paper proves that sequences of compact gradient Ricci solitons converge to orbifold solitons under certain geometric constraints, with special results in four dimensions linking curvature bounds to topological invariants.

Contribution

It establishes convergence results for compact Ricci solitons to orbifold limits under volume and curvature constraints, especially in four dimensions.

Findings

Convergence of Ricci solitons to orbifolds under specified conditions.

In dimension 4, $L^2$ curvature bounds relate to Euler number bounds.

Criteria for limits to be compact are provided.

Abstract

We show that sequences of compact gradient Ricci solitons converge to complete orbifold gradient solitons, assuming constraints on volume, the $L^{n/2}$-norm of curvature, and the auxiliary constant $C_1$. The strongest results are in dimension 4, where $L^2$ curvature bounds are equivalent to upper bounds on the Euler number. We obtain necessary and sufficient conditions for limits to be compact.