On Moduli Spaces of Ricci Solitons

Abstract

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 × C ∞ (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.

Stability and instability of Ricci solitons

“…The slice used in [PS13] is constructed via the exponential map of the weak Riemannian structure on M. Hence it differs from the affine sliceS g,fg that we use in this paper. However, we can identify the Ricci solitons in S g,fg andS g,fg via the map Ψ : P →S g,fg which associates to g 1 ∈ S g,fg the unique metricg 1 ∈S g,fg isometric to g 1 .…”

Rigidity and Infinitesimal Deformability of Ricci Solitons

“…Some obstructions against the existence of infinitesimal solitonic deformations are given in [PS13]. Analoguous questions have been studied in the Einstein case before [Koi78,Koi80,Koi82,Koi83], see also [Bes08] and the methods used here are quite similar.…”

“…Let M be the set of smooth metrics on M and assume that s ∈ N satisfies s ≥ [ n 2 ] + 3. According to [PS13], we introduce the following notions:…”

Rigidity of $SU_n$-type symmetric spaces

“…One can then ask whether a given Einstein metric g can be deformed through Ricci solitons. The structure theory of the moduli space of Ricci solitons was developed by Podesta and Spiro in [25] and we refer the reader to Section 5 for more details. All Hermitian symmetric spaces admit infinitesimal solitonic deformations.…”