# Seminars & Events for Joint Princeton Rutgers Geometric PDEs Seminar

##### Regularity Results for Optimal Transport Maps

Knowing whether optimal maps are smooth or not is an important step towards a qualitative understanding of them. In the 90's Caffarelli developed a regularity theory on R^n for the quadratic cost, which was then extended by Ma-Trudinger-Wang and Loeper to general cost functions which satisfy a suitable structural condition. Unfortunately, this condition is very restrictive, and when considered on Riemannian manifolds with the cost given by the squared distance, it is satisfied only in very particular cases. Hence the need to develop a partial regularity theory: is it true that optimal maps are always smooth outside a "small" singular set? The aim of this talk is to first review the "classical" regularity theory for optimal maps, and then describe some recent results about their partial regularity.

##### Hypersurfaces in hyperbolic space and conformal metrics on domains in sphere

In this talk I will introduce a global correspondence between properly immersed horospherically convex hyper surfaces in hyperbolic space and complete conformal metrics on subdomains in the boundary at infinity of

hyperbolic space. I will discuss when a horospherically convex hypersurface is proper, when its hyperbolic Gauss map is injective, and when it is embedded. These are expected to be useful to the understandings of both elliptic problems of Weingarten hypersurfaces in hyperbolic space and elliptic problems of complete conformal metrics on subdomains in sphere.

##### On the uniqueness of asymptotic limits of the Ricci flow

Given a compact Riemannian manifold we consider a solution of a normalization of the Ricci flow which exists for all time and such that both the full curvature tensor and the diameter of the manifold are uniformly bounded along the flow. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Analytic minimal model program with Ricci flow

I will introduce the analytic minimal model program proposed by Tian and myself to study formation of singularities of the Kahler-Ricci flow. We also construct geometric and analytic surgeries of codimension one and higher codimensions equivalent to birational transformations in algebraic geometry by Ricci flow.

##### Degenerations of Ricci Flat Kahler Metrics under extremal transitions and flops

We will discuss degeneration of Ricci-flat Kahler metrics on Calabi-Yau manifolds under algebraic geometric surgeries: extremal transitions or flops. We will prove a version of Candelas and de la Ossa's conjecture: Ricci-flat Calabi-Yau manifolds related via extremal transitions and flops can be connected by a path consisting of continuous families of Ricci-flat Calabi-Yau manifolds and a compact metric space in the Gromov-Hausdorff topology. This is joint work with Yuguang Zhang.

##### Quantitative rigidity estimates

For many classical rigidity questions in differential geometry it is natural to ask to which extent they are stable. I will review several recent results in the literature. A typical example is the following: there is a constant $C$ such that, if $\Sigma$ is a $2$-dimensional embedded closed surface in $R^3$, then $\min_\lambda \|A- \lambda g\|_{L^2} \leq C \|A - {\rm tr A} g/2\|_{L^2}$, where $A$ is the second fundamental form of the surface and $g$ the Riemannian metric as a submanifold of $R^3$.