The Soliton-Ricci Flow with variable volume forms

Nefton Pali (1)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay

We introduce a flow of Riemannian metrics and positive volume forms over compact oriented manifolds whose formal limit is a shrinking Ricci soliton. The case of a fixed volume form has been considered in our previous work. We still call this new flow, the Soliton-Ricci flow. It corresponds to a forward Ricci type flow up to a gauge transformation. This gauge is generated by the gradient of the density of the volumes. The new Soliton-Ricci flow exist for all times. It represents the gradient flow of Perelman's $\mathcal{W}$ functional with respect to a pseudo-Riemannian structure over the space of metrics and normalized positive volume forms. We obtain an expression of the Hessian of the $\mathcal{W}$ functional with respect to such structure. Our expression shows the elliptic nature of this operator in the orthogonal directions to the orbits obtained by the action of the group of diffeomorphism. In the case that initial data is K\"ahler, the Soliton-Ricci flow over a Fano manifold preserves the K\"ahler structure and the symplectic form. Over a Fano manifold, the space of tamed complex structures embeds naturally, via the Chern-Ricci map, to the space of metrics and normalized positive volume forms. Over such space the pseudo-Riemannian structure restricts to a Riemannian one. We perform a study of the sign of the restriction of the Hessian of the $\mathcal{W}$ functional over such space. This allows us to obtain a finite dimensional reduction of the stability problem for K\"ahler-Ricci solitons. This reduction represents the solution of this well known problem.

Mots clés: Kähler-Ricci solitons

Domaines: Variables complexes [math.CV]

Nefton Pali. The Soliton-Ricci Flow with variable volume forms. 2014. ⟨hal-01007701v3⟩