Nefton Pali

Variation formulas for the complex components of the Bakry-Emery-Ricci endomorphism

We compute first variation formulas for the complex components of the Bakry-Emery-Ricci endomorphism along Kähler structures. Our formulas show that the principal parts of the variations are quite standard complex differential operators with particular symmetry properties on the complex decomposition of the variation of the Kähler metric. We show as application that the Soliton-Kähler-Ricci flow generated by the Soliton-Ricci flow represents a complex strictly parabolic system of the complex components of the variation of the Kähler metric.

The Soliton-Ricci Flow with variable volume forms

Abstract 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 previouswork.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 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 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 Kähler, the Soliton-Ricci flow over a Fano manifold preserves the Kähler condition and the symplectic form. Over a Fano manifold, the space of tamed complex structures embeds naturally, via the Chern-Ricci map, into 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 W functional over such space. This allows us to obtain a finite dimensional reduction of the stability problem for Kähler-Ricci solitons. This reduction represents the solution of this well known problem. A less precise and less geometric version of this result has been obtained recently by the author in [28].

On the solutions of the Aubin equation and the K-energy of Einstein–Fano manifolds

We propose an improvement to the bifurcation technique considered by Bando–Mabuchi for the construction of the solutions of the Aubin equation over Einstein–Fano manifolds. We also introduce a simplification in Tians proof of the properness of the K-energy functional over Einstein–Fano manifolds with trivial holomorphic automorphisms group.

Concavity of Perelman’s \mathcal {W}-functional over the space of Kähler potentials

The concavity of Perelman’s

Degenerate complex Monge-Amp\`ere equations over compact K\"ahler manifolds

\mathcal {W}

The Soliton-Ricci Flow over Compact Manifolds

W-functional over a neighborhood of a Kähler–Ricci soliton inside the space of Kähler potentials is a direct consequence of author’s solution of the variational stability problem for Kähler–Ricci solitons. We provide a new and rather simple proof of this particular fact. This new proof uses in minor part some elementary formulas obtained in our previous work.

Variational stability of Kähler–Ricci solitons

We show an exact (i.e. no smooth error terms) Fourier inversion type formula for differential operators over Riemannian manifolds. This provides a coordinate free approach for the theory of pseudo-differential operators.