SGS XXIII: Sunday, November 10, 2013, Emory University

Abstract: We develop a gluing procedure designed to obtain canonical metrics on connected sums of Einstein four-manifolds. These metrics are critical points of quadratic Riemannian functionals. The main application is an existence result, using two well-known Einstein manifolds as building blocks: the Fubini-Study metric on CP2, and the product metric on S2 x S2.. Using these metrics in various gluing configurations, critical metrics are found on connected sums.

Abstract: Mean curvature flow is a nonlinear geometric flow where a hypersurface evolves to decrease its area as efficiently as possible. Singularities are unavoidable and the point is to understand the possible singularities and the behavior of the flow near a singularity. I will talk about recent joint work with Toby Colding and with Toby Colding and Tom Ilmanen where we prove strong rigidity and uniqueness theorems for generic singularities in all dimensions.

Abstract:We will consider compact Riemannian manifolds M with boundary N. We let IN be the unit vectors to M whose base point is on N and point inwards towards M. Similarly we define OUT. The scattering data (loosely speaking) of a Riemannian manifold with boundary is a map from IN to OUT which assigns to each unit vector V of IN a the unit vector W in OUT. W will be the tangent vector to the geodesic determined by V when that geodesic first hits the boundary N again. This may not be defined for all V since the geodesic might be trapped (i.e. never hits the boundary again). A manifold is said to be scattering rigid if for any Riemannian manifold Q with boundary isometric to N and with the same scattering data, then the boundary isometry extends to an isometry of M and Q. The lens data includes not only the scattering data but also the lengths of the geodesics. In this talk we will discuss some recent results on the scattering (and lens) rigidity problems. One thing we will discuss is recent work of my graduate student Haomin Weh on the relation between scattering data and lens data for surfaces. There are a number of manifolds that are known to be lens rigid and there are examples that are not scattering or lens rigid. All of the known examples of non-rigidity have trapped geodesics in them. In this talk we will see that the flat solid torus is scattering rigid. This is the first global scattering (or lens) rigidity result for a manifold that has a trapped geodesic. The main issue is to show that the unit vectors tangent to trapped geodesics in any such Q have measure 0 in the unit tangent bundle of Q. We will also consider scattering rigidity of a number of two dimensional manifolds (joint work with Pilar Herreros) which have trapped geodesics

Abstract:The logarithmic Minkowski problem asks for necessary and sufficient conditions such that a nonnegative finite Borel measure on the unit sphere is the cone-volume measure of a convex body in the Euclidean space. This problem and the classical Minkowski problem for surface area measures are two important cases of the $L_p$ Minkowski problem in the $L_p$ Brunn-Minkowski theory. The solution to the logarithmic Minkowski problem for the unit balls of finite dimensional Banach spaces (origin-symmetric convex bodies) is discussed. Its relation to geometric inequalities stronger than the Brunn-Minkowski inequality is explained.