Space-time constructions for the mean curvature flow in a Ricci flow background

TL;DR

This paper establishes a connection between mean curvature flow and Ricci flow through space-time constructions, linking Hamilton's differential Harnack estimates and relating geometric analysis to quantum gravity.

Contribution

It introduces a canonical soliton construction for mean curvature flow in Ricci flow backgrounds, revealing new links between geometric flows and functional estimates.

Findings

Link between mean curvature flow and Ricci flow via canonical solitons

Relation of the second fundamental form to Lott's modified F-functional

Connection between Harnack estimates for both flows

Abstract

Given a solution of the (backwards) Ricci flow one can construct a so called canonical soliton metric on space-time, introduced by E. Cabezas-Rivas and P. Topping. We observe that for a mean curvature flow within a (backwards) Ricci flow background, the space-time track of the mean curvature flow yields a canonical soliton of the coupled flow within the canonical Ricci soliton. We show that this provides a link between Hamilton's differential Harnack estimate for the mean curvature flow and Hamilton's differential Harnack estimate for the Ricci flow. Moreover the second fundamental form of our canonical soliton matches the boundary term of the evolution of J. Lott's modified F-functional for a Ricci flow with boundary. This functional also appears in quantum gravity.