Miles Simon

Ricci flow of non-collapsed three manifolds whose Ricci curvature is bounded from below

Abstract We consider complete (possibly non-compact) three dimensional Riemannian manifolds (M, g) such that: (a) (M, g) is non-collapsed (i.e. the volume of an arbitrary ball of radius one is bounded from below by v > 0), (b) the Ricci curvature of (M, g) is bounded from below by k, (c) the geometry at infinity of (M, g) is not too extreme (or (M, g) is compact). Given such initial data (M, g) we show that a Ricci flow exists for a short time interval 0, T), where T T(v, k) > 0. This enables us to construct a Ricci flow of any (possibly singular) metric space (X, d) which arises as a GromovHausdorff (GH) limit of a sequence of 3-manifolds which satisfy (a), (b) and (c) uniformly. As a corollary we show that such an X must be a manifold. This shows that the conjecture of M. AndersonJ. CheegerT. ColdingG. Tian is correct in dimension three.

Ricci flow of almost non-negatively curved three manifolds

Abstract In this paper we study the evolution of almost non-negatively curved (possibly singular) three dimensional metric spaces by Ricci flow. The non-negatively curved metric spaces which we consider arise as limits of smooth Riemannian manifolds (Mi, ig), i ∈ ℕ, whose Ricci curvature is bigger than –1/i, and whose diameter is less than d 0 (independent of i) and whose volume is bigger than v 0 > 0 (independent of i). We show for such spaces, that a solution to Ricci flow exists for a short time t ∈ (0, T), that the solution is smooth for t > 0, and has Ricci (g(t)) ≧ 0 and Riem (g(t)) ≧ c/t for t ∈ (0, T) (for some constant c = c(v 0, d 0, n)). This allows us to classify the topological type and the differential structure of the limit manifold (in view of the theorem of Hamilton [J. Diff. Geom. 24: 153–179, 1986] on closed three manifolds with non-negative Ricci curvature).

Local smoothing results for the Ricci flow in dimensions two and three

We present local estimates for solutions to the Ricci flow, without the assumption that the solution has bounded curvature. These estimates lead to a generalisation of one of the pseudolocality results of G.Perelman in dimension two.

Ricci Flow of Regions with Curvature Bounded Below in Dimension Three

We consider smooth complete solutions to Ricci flow with bounded curvature on manifolds without boundary in dimension three. Assuming an open ball at time zero of radius one has sectional curvature bounded from below by −1, then we prove estimates which show that compactly contained subregions of this ball will be smoothed out by the Ricci flow for a short but well-defined time interval. The estimates we obtain depend only on the initial volume of the ball and the distance from the compact region to the boundary of the initial ball. Versions of these estimates for balls of radius r follow using scaling arguments.

Some local estimates and a uniqueness result for the entire biharmonic heat equation

Abstract We consider smooth solutions to the biharmonic heat equation on ℝn × [0,T] for which the square of the Laplacian at time t is globally bounded from above by k0/t for some k0 in ℝ+, for all t ∈ [0,T]. We prove local, in space and time, estimates for such solutions. We explain how these estimates imply uniqueness of smooth solutions in this class.