On moduli spaces of Ricci solitons

TL;DR

This paper extends the deformation theory of Einstein metrics to shrinking Ricci solitons on compact manifolds, establishing the structure of their moduli spaces and criteria for rigidity.

Contribution

It introduces a framework for analyzing infinitesimal and local deformations of shrinking Ricci solitons, generalizing classical Einstein metric deformation theory.

Findings

Existence of a finite-dimensional submanifold containing the soliton pre-moduli space.

Definition of solitonic rigidity and criteria for it.

Analytic structure of the moduli space around a fixed Ricci soliton.

Abstract

We study deformations of shrinking Ricci solitons on a compact manifold M, generalising the classical theory of deformations of Einstein metrics. Using appropriate notions of twisted slices S_f inside the space of all Riemannian metrics on M, we define the infinitesimal solitonic deformations and the local solitonic pre-moduli spaces. We prove the existence of a finite dimensional submanifold of S_f x C^infty(M), which contains the pre-moduli space of solitons around a fixed shrinking Ricci soliton as an analytic subset. We define solitonic rigidity and give criteria which imply it.