An entropy formula for the heat equation on manifolds with time-dependent metric, application to ancient solutions

in Potential Analysis (2015), 42(2), 483-497

We introduce a new entropy functional for nonnegative solutions of the heat equation on a manifold with time-dependent Riemannian metric. Under certain integral assumptions, we show that this entropy is ... [more ▼]

We introduce a new entropy functional for nonnegative solutions of the heat equation on a manifold with time-dependent Riemannian metric. Under certain integral assumptions, we show that this entropy is non-decreasing, and moreover convex if the metric evolves under super Ricci flow (which includes Ricci flow and fixed metrics with nonnegative Ricci curvature). As applications, we classify nonnegative ancient solutions to the heat equation according to their entropies. In particular, we show that a nonnegative ancient solution whose entropy grows sublinearly on a manifold evolving under super Ricci flow must be constant. The assumption is sharp in the sense that there do exist nonconstant positive eternal solutions whose entropies grow exactly linearly in time. Some other results are also obtained. [less ▲]

Harnack estimates for geometric flows, applications to Ricci flow coupled with harmonic map flow

in Geometriae Dedicata (2014), 169(1), 411-418

We derive Harnack estimates for heat and conjugate heat equations in abstract geometric flows. The main results lead to new Harnack inequalities for a variety of geometric flows. In particular, Harnack ... [more ▼]

We derive Harnack estimates for heat and conjugate heat equations in abstract geometric flows. The main results lead to new Harnack inequalities for a variety of geometric flows. In particular, Harnack inequalities for the Ricci flow coupled with Harmonic map flow are obtained. [less ▲]

An entropy formula relating Hamiltonʼs surface entropy and Perelmanʼs W entropy

in Comptes Rendus. Mathématique (2013), 351(3-4), 115-118

In this note, based on Hamiltonʼs surface entropy formula, we construct an entropy formula of Perelmanʼs type for the Ricci flow on a closed surface with positive curvature. Similar to Perelmanʼs WW ... [more ▼]

In this note, based on Hamiltonʼs surface entropy formula, we construct an entropy formula of Perelmanʼs type for the Ricci flow on a closed surface with positive curvature. Similar to Perelmanʼs WW entropy, the critical point of our entropy is the gradient shrinking soliton; however, there is no conjugate heat equation involved. This shows a close relation between Hamiltonʼs entropy and Perelmanʼs W entropy on closed surfaces. [less ▲]