[Paper Review] Ricci flow on singular manifolds

Boris Vertman|arXiv (Cornell University)|Mar 21, 2016

Geometric Analysis and Curvature Flows11 references6 citations

TL;DR

This paper establishes local and long-time existence of Ricci flow on incomplete Riemannian manifolds with edge singularities and bounded Ricci curvature, proving bounded Riemann curvature tensor and regularity of the metric family. The analysis relies on the Lichnerowicz Laplacian and Ricci de Turck flow, extending results to spaces with isolated conical singularities when metrics are close to Ricci flat edge metrics.

ABSTRACT

In this paper we prove local existence of a Ricci flow starting at an incomplete manifold with edge singularities and bounded Ricci curvature, flowing for a short time within a class of incomplete edge manifolds. We derive regularity properties for the corresponding family of Riemannian metrics and establish boundedness of the Riemannian curvature tensor along the flow. For Riemannian metrics that are sufficiently close to a Ricci flat incomplete edge metric, we prove long time existence of the Ricci flow. Our results in particular include the case of spaces with isolated conical singularities. The proof works by a careful analysis of the Lichnerowicz Laplacian and the Ricci de Turck flow equation.

Motivation & Objective

• To establish the existence of Ricci flow on incomplete manifolds with edge singularities and bounded Ricci curvature.
• To analyze the regularity of the evolving Riemannian metrics and prove boundedness of the Riemann curvature tensor along the flow.
• To extend the existence result to long-time flow when the initial metric is sufficiently close to a Ricci flat incomplete edge metric.
• To include the case of isolated conical singularities as a special case of edge singularities.
• To provide a rigorous framework for geometric flows on singular spaces using the Ricci de Turck equation and Lichnerowicz Laplacian.

Proposed method

• Utilize the Ricci de Turck flow to transform the Ricci flow equation into a quasilinear parabolic system with a well-defined initial value problem.
• Analyze the Lichnerowicz Laplacian on incomplete edge manifolds to control the linearized Ricci operator and ensure solvability.
• Establish weighted Sobolev estimates for the linearized operator to handle the singular structure and maintain regularity.
• Apply maximal regularity theory in weighted Hölder spaces to prove short-time existence and smoothness of the flow.
• Use perturbation arguments to extend the flow beyond short time when the initial metric is close to a Ricci flat edge metric.
• Derive uniform bounds on the Riemann curvature tensor throughout the flow using the boundedness of Ricci curvature and parabolic regularity.

Experimental results

Research questions

• RQ1 Can Ricci flow be initiated on incomplete Riemannian manifolds with edge singularities and bounded Ricci curvature?
• RQ2 What regularity properties does the evolving Riemannian metric inherit under the Ricci flow in the presence of edge singularities?
• RQ3 Is the Riemann curvature tensor uniformly bounded along the Ricci flow on such singular manifolds?
• RQ4 Under what conditions does the Ricci flow exist for long time on incomplete edge manifolds?
• RQ5 How do conical singularities affect the existence and regularity of the Ricci flow, and can they be treated as a special case of edge singularities?

Key findings

• Local existence of Ricci flow is established for incomplete manifolds with edge singularities and bounded Ricci curvature, with the flow evolving within the class of incomplete edge manifolds.
• The Riemann curvature tensor remains uniformly bounded along the flow, ensuring controlled geometric evolution.
• The evolving Riemannian metrics exhibit regularity properties consistent with the underlying edge structure and initial curvature bounds.
• For initial metrics sufficiently close to a Ricci flat incomplete edge metric, the Ricci flow exists for all time.
• The results include the case of isolated conical singularities as a special instance of edge singularities.
• The analysis of the Lichnerowicz Laplacian and the Ricci de Turck equation provides the core technical foundation for all existence and regularity results.