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Pré-Publication, Document De Travail Année : 2014

The Soliton-Ricci Flow with variable volume forms

Nefton Pali
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Résumé

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 generated by the gradient of the density of the volumes. The new Soliton-Ricci flow exist for all times and 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 directions orthogonal to the orbits obtained by the action of the group of diffeomorphism. In the case the initial data is Kähler then the Soliton-Ricci flow preserves the Kähler structure and the symplectic form. The space of tamed complex structures embeds naturally to the space of metrics and normalized positive volume forms via the Chern-Ricci map. 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, and thus the solution, of the well known problem of the stability of Kähler-Ricci solitons.

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Dates et versions

hal-01007701 , version 1 (17-06-2014)
hal-01007701 , version 2 (19-11-2014)
hal-01007701 , version 3 (14-01-2015)

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  • HAL Id : hal-01007701 , version 1

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Nefton Pali. The Soliton-Ricci Flow with variable volume forms. 2014. ⟨hal-01007701v1⟩
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