Ricci DeTurck flow on incomplete manifolds

Abstract

In this paper we construct a Ricci DeTurck 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 W.-X. Shi for complete manifolds [J. Differ. Geom. 30, No. 1, 223–301 (1989; Zbl 0676.53044)], this gives that any (complete or incomplete) manifold of bounded curvature can be evolved by the Ricci DeTurck flow for a short time.