An energy approach to the problem of uniqueness for the Ricci flow

Abstract

Abstract. We revisit the problem of uniqueness for the Ricci flow and give a short, direct proof, based on the consideration of a simple energy quantity, of Hamilton/Chen-Zhu's theorem on the uniqueness of complete solutions of uniformly bounded curvature. With a variation of this quantity and technique, we further prove a uniqueness theorem for subsolutions to a general class of mixed differential inequalities which implies an extension of Chen-Zhu's result to solutions (and initial data) of potentially unbounded curvature.Let M = M n be a smooth manifold and g 0 a Riemannian metric on M . In this paper, we revisit the question of uniqueness of solutions to the initial value problemassociated to the Ricci flow on M . The broadest category in which uniqueness is currently known to hold without dimensional restrictions is that of complete solutions of uniformly bounded curvature.Theorem 1 (Hamilton [H1]; Chen-Zhu [CZ]). Suppose g 0 is a complete metric and g(t) andg(t) are solutions to the initial value problem (1) satisfyingThe uniqueness of solutions to (1) is not an automatic consequence of the theory of parabolic equations, since the Ricci flow equation is only weakly-parabolic. For compact M , there are two basic arguments, both due to Hamilton. The first appears in Hamilton's orginal paper [H1] as a byproduct of the proof of the short-time existence of solutions and is based on a Nash-Moser-type inverse function theorem. The second, given in [H2], effectively reduces the question of uniqueness to that for the strictly parabolic Ricci-DeTurck flow. The basis of this argument is the observation that the DeTurck diffeomorphisms, which are generally obtained as solutions to a system of ODE depending on a given solution to the Ricci-DeTurck flow, can also be represented as the solutions to a certain parabolic PDE -a harmonic map heat flow -which depends on the associated solution to the Ricci flow. As DeTurck's method is applicable to many other geometric evolution equations with gauge-based degeneracies, this second argument of Hamilton's gives rise to an elegant and flexible general prescription in which one exchanges the problem of uniqueness for one weakly parabolic system for the (separate) problems of existence and uniqueness for one or more auxiliary strictly parabolic systems.

Uniqueness of Schrödinger flow on manifolds

“…The energy method can also be used in proving uniqueness of other types of geometric flows. For example, Kotschwar applied the energy method to prove the uniqueness of Ricci flow [13,14]. Part of our motivation of the current work arises from our study of another Schrödinger type geometric flow, namely, the Skew Mean Curvature Flow(SMCF) [20].…”

Second-Order Geometric Flows on Foliated Manifolds

“…The estimates are proved in [15] for the Ricci flow in the non-foliated case. Since all the estimates in [15] are local and a solution of the transverse Ricci flow can be regarded as a collection of solutions to the Ricci flow on open sets in R n , the claim follows. …”

“…In this section we give a proof of the well-posedness of the of the transverse Ricci flow based on theorem 1.1: the short-time existence is treated in the spirit of [8], while the uniqueness is obtained with the energy approach of [15].…”

“…In order to prove the uniqueness of the transverse Ricci flow, we adapt the argument of [15] to the compact foliated case. Assume that g Q andḡ Q are two solutions in the interval [0, T ] of the transverse Ricci-flow with the same initial valueg Q .…”

“…The modified transverse Ricci flow is strongly transversally parabolic and it is well-posed in view of theorem 1.1. The existence of a solution to the transverse Ricci flow follows from the wellposedness of its modification, while for the uniqueness of the solutions we adapt an argument in [15] to the foliated case. In this second part we have to assume that the foliation is homologically orientable in order to introduce an integral functional.…”