Hamilton-Jacobi Equations and Distance Functions on Riemannian Manifolds

The paper is concerned with the properties of the distance function from a closed subset of a Riemannian manifold, with particular attention to the set of singularities

A Flow Tangent to the Ricci Flow via Heat Kernels and Mass Transport

We present a new relation between the short time behavior of the heat flow, the geometry of optimal transport and the Ricci flow. We also show how this relation can be used to define an evolution of metrics on non-smooth metric measure spaces with Ricci curvature bounded from below

Some remarks on Huisken's monotonicity formula for mean curvature flow

We discuss a monotone quantity related to Huisken's monotonicity formula and some technical consequences for mean curvature flow.Comment: in "Singularities in nonlinear evolution phenomena and applications", 157-169, CRM Series, 9, Ed. Sc. Norm. Pisa, 200

Evolution of the Weyl Tensor under the Ricci Flow

We compute the evolution equation of the Weyl tensor under the Ricci flow of a Riemannian manifold and we discuss some consequences for the classification of locally conformally flat Ricci solitons

Locally conformally flat ancient Ricci flows

We show that any locally conformally flat ancient solution to the Ricci flow must be rotationally symmetric. As a by-product, we prove that any locally conformally flat Ricci soliton is a gradient soliton in the shrinking and steady cases as well as in the expanding case, provided the soliton has nonnegative curvature.Comment: Final version, to appear on Anal. PD

Motion by curvature of networks with two triple junctions

We consider the evolution by curvature of a general embedded network with two triple junctions. We classify the possible singularities and we discuss the long time existence of the evolution