Bounded Ricci curvature and positive scalar curvature under Ricci flow

Klaus Kröncke, Tobias Marxen and Boris Vertman

Vol. 324 (2023), No. 2, 295–331

DOI: 10.2140/pjm.2023.324.295

Abstract

We consider a Ricci de Turck flow of spaces with isolated conical singularities, which preserves the conical structure along the flow. We establish that a given initial regularity of Ricci curvature is preserved along the flow. Moreover under additional assumptions, positivity of scalar curvature is preserved under such a flow, mirroring the standard property of Ricci flow on compact manifolds. The analytic difficulty is the a priori low regularity of scalar curvature at the conical tip along the flow, so that the maximum principle does not apply. We view this work as a first step toward studying positivity of the curvature operator along the singular Ricci flow.

Keywords

Ricci flow, positive scalar curvature, conical singularities

Mathematical Subject Classification

Primary: 53E20

Secondary: 53C25, 58J35

Milestones

Received: 4 May 2022

Revised: 6 May 2023

Accepted: 13 May 2023

Published: 26 July 2023