We consider the evolution of a closed convex hypersurface under a volume preserving curvature flow. The speed is given by a power of the mth mean curvature plus a volume preserving term, including the case of powers of the mean curvature or of the Gauss curvature. We prove that if the initial hypersurface satisfies a suitable pinching condition, the solution exists for all times and converges to a round sphere. Mathematics Subject Classification (2000)53C44 · 35K55 · 58J35 · 35B40

We generalize most of the known Ricci flow invariant non-negative curvature conditions to less restrictive negative bounds that remain sufficiently controlled for a short time.As an illustration of the contents of the paper, we prove that metrics whose curvature operator has eigenvalues greater than −1 can be evolved by the Ricci flow for some uniform time such that the eigenvalues of the curvature operator remain greater than −C. Here the time of existence and the constant C only depend on the dimension and the degree of non-collapsedness. We obtain similar generalizations for other invariant curvature conditions, including positive biholomorphic curvature in the Kähler case. We also get a local version of the main theorem.As an application of our almost preservation results we deduce a variety of gap and smoothing results of independent interest, including a classification for non-collapsed manifolds with almost non-negative curvature operator and a smoothing result for singular spaces coming from sequences of manifolds with lower curvature bounds. We also obtain a short-time existence result for the Ricci flow on open manifolds with almost non-negative curvature (without requiring upper curvature bounds). arXiv:1707.03002v2 [math.DG] 25 Jul 2017 for some ε ∈ [0, 1]. Then the Ricci flow g(t) with initial metric g exists until time τ , is Kähler if (M, g) is Kähler, and we have the curvature bounds Rm g(t) + Cε I ∈ C and |Rm g(t) | ≤ C t for all t ∈ (0, τ ].(1.2)

We prove short time existence for the Ricci flow on open manifolds of non-negative complex sectional curvature without requiring upper curvature bounds. By considering the doubling of convex sets contained in a Cheeger–Gromoll convex exhaustion and solving the singular initial value problem for the Ricci flow on these closed manifolds, we obtain a sequence of closed solutions of the Ricci flow with non-negative complex sectional curvature which subconverge to a Ricci flow on the open manifold. Furthermore, we find an optimal volume growth condition which guarantees long time existence, and give an analysis of the long time behavior of the Ricci flow. We also construct an explicit example of an immortal non-negatively curved Ricci flow with unbounded curvature for all time.

In an ambient space with rotational symmetry around an axis (which include the Hyperbolic and Euclidean spaces), we study the evolution under the volume-preserving mean curvature flow of a revolution hypersurface M generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces (tgh for short). Requiring that, for each time t ≥ 0, the evolving hypersurface M t meets such tgh ortogonally, we prove that: a) the flow exists while M t does not touch the axis of rotation; b) throughout the time interval of existence, b1) the generating curve of M t remains a graph, and b2) the averaged mean curvature is double side bounded by positive constants; c) the singularity set (if non-empty) is finite and lies on the axis; d) under a suitable hypothesis relating the enclosed volume to the n-volume of M , we achieve long time existence and convergence to a revolution hypersurface of constant mean curvature.