Geometric Analysis and Riemannian geometry (G. Catino, G. Grillo, D. Monticelli, M. MuratoriF. Punzo)

The fundamental problem of capturing the topological properties of a manifold by its metric structure opened, in the last decades, extremely fruitful areas of mathematics. From this perspective, there has been an increasing interest in the study of Riemannian manifolds endowed with metrics satisfying special structural equations, possibly involving curvatures and vector fields. These structures arise naturally not only in several different mathematical frameworks, but also in Physics, in particular in General Relativity. Some directions of research are:

(1) Geometric and analytic properties of well known special Riemannian structures and related flows such as: Ricci solitons/Ricci flow and Yamabe solitons/Yamabe flow.

(2) New structures and new geometric flows.  On one hand, starting from promising recent results, our purpose is to study Riemannian metrics satisfying general structural conditions, involving their Ricci curvature, scalar curvature and globally defined vector fields, that can be considered as a perturbation of the Einstein equation and that include, as particular cases, most of the important examples considered recently in the literature. For instance, we explored geometric and analytic features of what we have called "Einstein-type manifolds'', both in the gradient and in the more difficult non-gradient case. 

On the other hand, we are investigating new geometric flows (for instance the Ricci-Bourguignon flow and the Renormalization Group flow) which have been very recently introduced and promise to be useful tools in the study of the geometry of a Riemannian manifold. In analogy with the Ricci soliton case, most of these Einstein-type manifolds arise as self-similar solutions and possible singularity models of the corresponding flows.

(3) Riemannian curvature functionals and critical metrics. It is well known that Einstein metrics can be found as critical points of the Einstein-Hilbert functional (with constrained volume). We are studying Riemannian metrics which are solutions of the Euler-Lagrange equations associated to more general curvature functionals, with particular attention to the quadratic case. Due to the complexity of the variational PDEs involved, it is not clear which conditions can guarantee rigidity results, i.e non-existence of non-trivial (non-Einstein) solutions.

(4) Classification of Einstein Spacetime solutions under curvature conditions.

(5) Liouville type result for elliptic and parabolic equations on Riemannian manifolds.

Politecnico di Milano - Dipartimento di Matematica