Ricci de Turck Flow on Singular Manifolds

Abstract

In this paper we prove local existence of a Ricci de Turck flow starting at a space with incomplete edge singularities and flowing for a short time within a class of incomplete edge manifolds. We derive regularity properties for the corresponding family of Riemannian metrics and discuss boundedness of the Ricci curvature along the flow. For Riemannian metrics that are sufficiently close to a flat incomplete edge metric, we prove long time existence of the Ricci de Turck flow. Under certain conditions, our results yield existence of Ricci flow on spaces with incomplete edge singularities. The proof works by a careful analysis of the Lichnerowicz Laplacian and the Ricci de Turck flow equation.

Stability of Ricci de Turck flow on singular spaces

“…In the present discussion, which can be viewed as a continuation of the recent work by the second named author in [Ver16], we consider Ricci de Turck flow preserving isolated conical singularities and establish a stability result near Ricci-flat metrics. The crucial difficulty in our setting is in addition to the singularity of the underlying space the tensorial nature of the flow, in contrast to the two-dimensional or the Kähler setting.…”

“…We now proceed as follows. We first recall geometric aspects of isolated conical singularities and define Hölder spaces adapted to the singular geometry and mapping properties of the heat kernel as in [Ver16]. We then conclude the introduction with statement of the main results.…”

“…After a discussion of the Lichnerowicz Laplacian and its Friedrichs extension in §2, we proceed with a detailed characterization of tangentially stable Einstein manifolds in §3. In §5 we review the mapping properties of the heat operator as established in [Ver16]. In §6 we establish exponential large time estimates for the heat operator norms.…”

“…Here, the rough Laplacian ∆ = ∇ * ∇ is defined with the sign convention such that its eigenvalues are non-negative and the Riemannian curvature tensor is used with the sign convention such thatRh = h. We choose local coordinates (x, z) over the singular neighborhood C(F) = (0, 1) x × F. In the previous paper [Ver16] we have introduced a decomposition of compactly supported smooth sections…”

“…This section is basically a recap of the corresponding definitions in [Ver16] in the case of isolated conical singularities. We introduce Hölder spaces adapted to the singular geometry and mapping properties of the corresponding heat operator.…”