Ricci de Turck flow on incomplete manifolds

TL;DR

This paper constructs a Ricci de Turck flow on incomplete manifolds with bounded curvature, preserving initial singularities and extending the flow's applicability to incomplete cases.

Contribution

It introduces a method to evolve incomplete manifolds with bounded curvature using Ricci de Turck flow, maintaining their initial singularity structure.

Findings

Flow remains uniformly equivalent to initial metric

Applicable to both complete and incomplete manifolds

Preserves initial singularity structures

Abstract

In this paper we construct a Ricci de Turck flow on any incomplete Riemannian manifold with bounded curvature. The central property of the flow is that it stays uniformly equivalent to the initial incomplete Riemannian metric, and in that sense preserves any given initial singularity structure. Together with the corresponding result by Shi for complete manifolds, this gives that any (complete or incomplete) manifold of bounded curvature can be evolved by the Ricci de Turck flow for a short time.