Venue/Program

Title: Geometric heat flows on conic spaces

Abstract: It has emerged in the past decade that one reason that nonlinear heat flows with singular initial data are particularly delicate is because in many cases these are ill-posed problems. I will talk about this in two `geometrically regular’ settings: the Ricci flow on two dimensional spaces with conic singularities, and the curvature flow on networks of curves. In both cases I’ll describe some of the delicate issues in establishing local existence theorems, the resolutions of which lead to sharp existence in a certain class of spaces with accompanying sharp regularity statements. The Ricci flow result is old joint work with Rubinstein and Sesum, and the network flow is joint with Lira, Pluda and Saez.

Title: Existence and uniqueness for crystalline mean curvature flows

Abstract:

In this seminar we will discuss some weak formulations for crystalline mean curvature flows, recently introduced in collaboration with A. Chambolle, M. Morini and M. Novaga. In particular we show that the Almgren-Taylor-Wang scheme starting from any given initial set converges, up to fattening, to a unique flat flow. For a special class of regular mobilities we show that the flat flow coincides with the unique distributional solution, according with our weak formulation.

Title: On anisotropic curvature flow

Abstract:

In this talk I will discuss anisotropic curvature motion for planar immersed curves and give a short-time existence result that holds for general anisotropies. This is joint work with G. Mercier and M. Novaga

Title: Optimal isoperimetric inequalities for surfaces in any codimension in Cartan-Hadamard manifolds

Abstract:

Let (M^n,g) be simply connected, complete, with non-positive sectional curvatures, and \Sigma a 2-dimensional surface in M^n. Let S be an area minimising 3-current such that \partial S = \Sigma. We use a weak mean curvature flow, obtained via elliptic regularisation, starting from \Sigma, to show that S satisfies the optimal Euclidean isoperimetric inequality: |S| \leq 1/(6\sqrt{\pi}) |\Sigma|^{3/2}. We also obtain the optimal estimate in case the sectional curvatures of M are bounded from above by -\kappa < 0 and characterise the case of equality. The proof follows from an almost monotonicity of a suitable isoperimetric difference along the approximating flows in one dimension higher.

Title: Variational and nonlocal curvature flows

Abstract: We will describe some general existence/uniqueness results for flows defined by translation-invariant "curvatures" satisfying a few basic axioms. This can be applied to variational flows, that is, flows derived as "gradient flows" of some perimeters, as well as nonlinear variants of these flows.

Title: Local Ricci flow and limits of non-collapsed regions whose Ricci curvature is bounded from below

Abstract:

We use a local Ricci flow to obtain a bi-Hölder correspondence between non-collapsed (possibly non-complete) 3-manifolds with Ricci curvature bounded from below and Gromov-Hausdorff limits of sequences thereof. This is joint work with Peter Topping and the proofs build on results and ideas from recent papers of Hochard and Topping+Simon.

Title: On Chen’s flow

Abstract: Chen’s operator for a submanifold is the twice iterated Laplacian on the pullback bundle, sometimes known as the rough Laplacian in the literature. Chen’s conjecture is that if Chen’s operator applied to the immersion map vanishes, then the submanifold is minimal. In the last few years, work has progressed on the parabolic flow with velocity corresponding to Chen’s operator applied to the immersion. Algebraically this flow sits close to surface diffusion and Willmore flow, but qualitatively its behaviour is much closer to the mean curvature flow. In particular, spheres shrink to points in finite time. In this talk we describe some recent work on Chen’s flow in two and four dimensions.

Title: Homogenization of parabolic equations

Abstract:

We consider the homogenization of a semilinear parabolic equation with vanishing viscosity and with an oscillating  potential. According to the rate between the frequency of oscillations in the potential and the vanishing viscosity factor, we obtain different regimes in the limit evolution and we discuss the locally uniform convergence of the solutions to the effective problem. It turns out that in the strong diffusion regime the effective operator is discontinuous in the gradient entry. We discuss the main properties of the solutions to the effective problem, and we show uniqueness for some classes of initial data.

Title: Pyramid Ricci flows