The (logarithmic) Sobolev inequalities along geometric flow and applications

TL;DR

This paper establishes (logarithmic) Sobolev inequalities for various geometric flows, leading to insights on long-term geometric properties and extending known results to new flow types and settings.

Contribution

It generalizes Sobolev inequalities and geometric properties to a broader class of flows, including mean curvature flow in Lorentzian space and twisted Kähler-Ricci flow.

Findings

Derived (logarithmic) Sobolev inequalities for multiple geometric flows

Proved long-time non-collapsing and non-inflated properties for these flows

Extended results to mean curvature flow in Lorentzian space and twisted Kähler-Ricci flow

Abstract

For some class of geometric flows, we obtain the (logarithmic) Sobolev inequalities and their equivalence up to different factors directly and also obtain the long time non-collapsing and non-inflated properties, which generalize the results in the case of Ricci flow or List-Ricci flow or harmonic-Ricci flow. As applications, for mean curvature flow in Lorentzian space with nonnegative sectional curvature and twisted K\"ahler-Ricci flow on Fano manifolds, we get the results above.