Smoothing and non-smoothing via a flow tangent to the Ricci flow

TL;DR

The paper investigates a transformation of metric measure spaces that replaces the original distance with a heat kernel measure-based length distance, demonstrating its smoothing effects on certain singular spaces and its ability to instantaneously regularize sub-Riemannian manifolds.

Contribution

It extends the understanding of heat kernel-based transformations by analyzing their smoothing effects on Euclidean cones and the Heisenberg group, showing instant regularization to Riemannian manifolds.

Findings

Singularity persists at the apex of some Euclidean cones after transformation.

The transformation instantaneously regularizes the Heisenberg group to a smooth Riemannian manifold.

Abstract

We study a transformation of metric measure spaces introduced by Gigli and Mantegazza consisting in replacing the original distance with the length distance induced by the transport distance between heat kernel measures. We study the smoothing effect of this procedure in two important examples. Firstly, we show that in the case of some Euclidean cones, a singularity persists at the apex. Secondly, we generalize the construction to a sub-Riemannian manifold, namely the Heisenberg group, and show that it regularizes the space instantaneously to a smooth Riemannian manifold.