Convex ancient solutions of the mean curvature flow

We study solutions of the mean curvature flow which are defined for all negative curvature times, usually called ancient solutions. We give various conditions ensuring that a closed convex ancient solution is a shrinking sphere. Examples of such conditions are: a uniform pinching condition on the curvatures, a suitable growth bound on the diameter or a reverse isoperimetric inequality. We also study the behaviour of uniformly k-convex solutions, and consider generalizations to ancient solutions immersed in a sphere

Ancient solutions to the Ricci flow with pinched curvature

We show that any ancient solution to the Ricci flow which satisfies a suitable curvature pinching condition must have constant sectional curvature.Comment: to appear in Duke Math Journa

Mean curvature flow singularities for mean convex surfaces

We study the evolution by mean curvature of a smooth n–dimensional surfaceM Rn+1, compact and with positive mean curvature. We first prove an estimate on the negative part of the scalar curvature of the surface. Then we apply this result to study the formation of singularities by rescaling techniques, showing that there exists a sequence of rescaled flows converging to a smooth limit flow of surfaces with nonnegative scalar curvature. This gives a classification of the possible singular behaviour for mean convex surfaces in the case n = 2

A note on maximal estimates for stochastic convolutions

In stochastic partial differential equations it is important to have pathwise regularity properties of stochastic convolutions. In this note we present a new sufficient condition for the pathwise continuity of stochastic convolutions in Banach spaces.Comment: Minor correction

Semiconcavity of the value function for exit time problems with nonsmooth target

We prove semiconcavity of the value function of a nonlinear optimal control problem where the cost functional depends on the arrival time of the trajectory on a given target set. We make suitable smoothness assumptions on the dynamics of the system, while the target set can be completely general. As a corollary, we prove differentiability of the value function for a class of linear systems

Convex hypersurfaces evolving by volume preserving curvature flows

We consider the evolution of a closed convex hypersurface in euclidean space under a volume preserving flow whose speed is given by a positive power of the mean curvature. We prove that the solution exists for all times and converges to a sphere. The result does not assume the curvature pinching properties or the restrictions on the dimension that were usually required in the previous literature. The proof of the convergence exploits the monotonicity of the isoperimetric ratio satisfied by this class of flows

Regularity along optimal trajectories of the value function of a Mayer problem

We consider an optimal control problem of Mayer type and prove that, under suitable conditions on the system, the value function is differentiable along optimal trajectories, except possibly at the endpoints. We provide counterexamples to show that this property may fail to hold if some of our conditions are violated. We then apply our regularity result to derive optimality conditions for the trajectories of the system

Convexity estimates for mean curvature flow and singularities of mean convex surfaces

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