In this paper, we prove a unique continuation or "backwards-uniqueness" theorem for solutions to the Ricci flow. A particular consequence is that the isometry group of a solution cannot expand within the lifetime of the solution.

In the first part of this paper, we prove local interior and boundary gradient estimates for p-harmonic functions on general Riemannian manifolds. With these estimates, following the strategy in recent work of R. Moser, we prove an existence theorem for weak solutions to the level set formulation of the 1/H (inverse mean curvature) flow for hypersurfaces in ambient manifolds satisfying a sharp volume growth assumption. In the second part of this paper, we consider two parabolic analogues of the p-harmonic equation and prove sharp Li-Yau type gradient estimates for positive solutions to these equations on manifolds of nonnegative Ricci curvature. For one of these equations, we also prove an entropy monotonicity formula generalizing an earlier such formula of the second author for the linear heat equation. As an application of this formula, we show that a complete Riemannian manifold with nonnegative Ricci curvature and sharp L p -logarithmic Sobolev inequality must be isometric to Euclidean space.

We show that if a shrinking soliton is asymptotic to a cone along an end then the isometry group of the cross-section of the cone embeds in the isometry group of the end of the shrinker. We also provide sufficient conditions for the isometries of the end to extend to the entire shrinker.

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.

We show that the norm of the Riemann curvature tensor of any smooth solution to the Ricci flow can be explicitly estimated in terms of its initial values on a given ball, a local uniform bound on the Ricci tensor, and the elapsed time. This provides a new, direct proof of a result ofŠesǔm, which asserts that the curvature of a solution on a compact manifold cannot blow up while the Ricci curvature remains bounded, and extends its conclusions to the noncompact setting. We also prove that the Ricci curvature must blow up at least linearly along a subsequence at a finite time singularity.