Geometric singularities and a flow tangent to the Ricci flow

TL;DR

This paper studies a geometric flow related to Ricci flow on spaces with singularities, establishing regularity and distance equivalence results for rough metrics and demonstrating the flow's behavior near conical singularities.

Contribution

It extends the Gigli-Mantegazza flow to spaces with geometric singularities, providing regularity results and showing the flow preserves smoothness away from singularities.

Findings

The flow's regularity depends on heat kernel regularity.

Distance induced by the flow matches the Gigli-Mantegazza flow on RCD(K,N) spaces.

Smooth manifolds with conical singularities remain smooth away from singularities over time.

Abstract

We consider a geometric flow introduced by Gigli and Mantegazza which, in the case of smooth compact manifolds with smooth metrics, is tangen- tial to the Ricci flow almost-everywhere along geodesics. To study spaces with geometric singularities, we consider this flow in the context of smooth manifolds with rough metrics with sufficiently regular heat kernels. On an appropriate non- singular open region, we provide a family of metric tensors evolving in time and provide a regularity theory for this flow in terms of the regularity of the heat kernel. When the rough metric induces a metric measure space satisfying a Riemannian Curvature Dimension condition, we demonstrate that the distance induced by the flow is identical to the evolving distance metric defined by Gigli and Mantegazza on appropriate admissible points. Consequently, we demonstrate that a smooth compact manifold with a finite number of geometric conical singularities remains a smooth manifold with a smooth metric away from the cone points for all future times. Moreover, we show that the distance induced by the evolving metric tensor agrees with the flow of RCD(K, N) spaces defined by Gigli-Mantegazza.