The purpose of this paper is to evolve non-smooth Riemannian metric tensors by the dual Ricci-Harmonic map flow. This flow is equivalent (up to a diffeomorphism) to the Ricci flow. One application will be the evolution of metrics which arise in the study of spaces whose curvature is bounded from above and below in the sense of Aleksandrov, and whose curvature operator (in dimension three Ricci curvature) is non-negative. We show that such metrics may always be deformed to a smooth metric having the same properties in a strong sense.
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 Gromov-Hausdorff 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.Anderson-J.Cheeger-T.Colding-G.Tian is correct in dimension three.* part of this work was completed during the author's stay at Universität Münster in the semester 2008/09. This work was partially supported by SFB/Transregio 71.
Abstract. We prove uniform curvature estimates for homogeneous Ricci flows: For a solution defined on [0, t] the norm of the curvature tensor at time t is bounded by the maximum of C(n)/t and C(n)(scal(g(t)) − scal(g(0))). This is used to show that solutions with finite extinction time are Type I, immortal solutions are Type III and ancient solutions are Type I, where all the constants involved depend only on the dimension n. A further consequence is that a non-collapsed homogeneous ancient solution on a compact homogeneous space emerges from a unique Einstein metric on the same space.The above curvature estimates are proved using a gap theorem for Ricciflatness on homogeneous spaces. The proof of this gap theorem is by contradiction and uses a local W 2,p convergence result, which holds without symmetry assumptions.The proof of Thurston's geometrization conjecture by Perelman [Per1], [Per2], [Per3] using Hamilton's Ricci flow [Ha] can certainly be considered a major break through. There are however interesting related problems which remain open. For instance, Lott asked in [Lo2], whether the 3-dimensional Ricci flow detects the homogeneous pieces in the geometric decomposition proposed by Thurston. In the same paper this was proved to be true for immortal solutions, assuming a Type III behavior of the curvature tensor and a natural bound on the diameter of the underlying closed oriented manifold. More recently, Bamler showed in a series of papers that for the Ricci flow with surgery there exist only finitely many surgery times, and that the Type III behavior holds after the last surgery time. In many cases, convergence to a geometric piece could be established. We refer to [Bam] and the papers quoted therein.Recall that a Ricci flow solution is called homogeneous, if it is homogeneous at every time. In dimension 3 homogeneous Ricci flows are well understood: see [IJ]
Abstract. We consider Ricci flow of complete Riemannian manifolds which have bounded non-negative curvature operator, non-zero asymptotic volume ratio and no boundary. We prove scale invariant estimates for these solutions. Using these estimates, we show that there is a limit solution, obtained by scaling down this solution at a fixed point in space. This limit solution is an expanding soliton coming out of the asymptotic cone at infinity.
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