Exact Fourier inversion formula over manifolds

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.Comment: No figure

Concavity of Perelman's W\mathcal{W}-functional over the space of K\"ahler potentials

In this short note we observe that the concavity of Perelman's $\mathcal{W}$-functional over a neighborhood of a K\"ahler-Ricci soliton inside the space of K\"ahler potentials is a direct consequence of author's solution of the variational stability problem for K\"ahler-Ricci solitons. Independently, we provide a rather simple proof of this fact based on some elementary formulas obtained in our previous work

Chern-Ricci invariance along G-geodesics

Over a compact oriented manifold, the space of Riemannian metrics and normalised positive volume forms admits a natural pseudo-Riemannian metric $G$, which is useful for the study of Perelman's $\mathcal{W}$ functional. We show that if the initial speed of a $G$-geodesic is $G$-orthogonal to the tangent space to the orbit of the initial point, under the action of the diffeomorphism group, then this property is preserved along all points of the $G$-geodesic. We show also that this property implies preservation of the Chern-Ricci form along such $G$-geodesics, under the extra assumption of complex aniti-invariant initial metric variation and vanishing of the Nijenhuis tensor along the $G$-geodesic. This result is useful for a slice type theorem needed for the proof of the dynamical stability of the Soliton-K\"ahler-Ricci flow

On maximally totally real embeddings

We consider complex structures with totally real zero section of the tangent bundle. We assume that the complex structure tensor is real-analytic along the fibers of the tangent bundle. This assumption is quite natural in view of a well known result by Bruhart and Whitney. We provide explicit integrability equations for such complex structures in terms of the fiberwise Taylor expansion.Comment: 56 pages, no figure