# Seminars & Events for Differential Geometry & Geometric Analysis Seminar

##### Adams and Moser-Trudinger inequalities: from concrete to abstract, and back

In this talk I will briefly review Moser-Trudinger inequalities, and David Adams' important contributions to the subject. I will then talk about recent work with Luigi Fontana, where we extend, unify and improve Adams' results in an abstract, measure-theoretic setting. I will then present a few of the several applications of such results, including: sharp inequalities for general elliptic differential operators, sharp inequalities on the Heisenberg group and the CR sphere, the Beckner-Onofri inequality (in joint work with Branson and Fontana) and sharp inequalities without boundary conditions on the unit ball of R^n.

##### A Paneitz-type operator for CR pluriharmonic functions

Paul Yang and I recently found a new fourth order CR-invariant operator on CR pluriharmonic functions in three dimensions which generalizes to the abstract setting the operator on the sphere discovered by Branson, Fontana, and Morpurgo. I will describe the similarities between this operator and the (conformal) Paneitz operator, and how this helps to better understand some problems in CR geometry.

##### positivity questions in K\"ahler-Einstein theory

In this talk, I describe how the existence of both complete and incomplete K"ahler-Einstein(KE) metrics on quasi-projective varieties motivates a number of questions around the theme of positivity in complex geometry. After describing in some details the case of complex surfaces, I discuss some higher dimensional results. Finally, I describe some applications not directly connected with the theory of singular KE metrics. For example, I show how to count the number of maximal parabolic subgroups in neat arithmetic lattices of the Bergman ball.

##### V-soliton equations, symplectic reductions and Kahler-Ricci flow

It has been widely recognized by now that in order to understand the geometry of Kahler manifolds under the Ricci flow one needs to understand a geometric-analytic version of Mori's Minimal Model Program. One of the major stumbling blocks is due to formation of finite time singularities. In recent work, Tian and I proposed an approach to the description of finite time singulariteis which relates the flow and its singularity formation to variation of symplectic reductions of a Kaehler manifold endowed with a 1-dimensional (complex) Hamiltonian torus action, where the Kaehler metric in the total space satisfies a static elliptic equation of soliton type (called V-soliton). I will explain how this works and how it relates to a Geometric version of the Minimal Model program.

##### Densities in Geometry, Including Isoperimetric Problems and the Poincaré Conjecture

Densities or weights play an important role throughout geometry, including Perelman's proof of the Poincaré Conjecture, Lawlor's new, elegant proof of the Double Bubble Theorem, and the isoperimetric problem.

##### On the degeneracy of optimal transport

It is a well known result of Caffarelli that an upper and lower bound on the Monge-Amp{\`e}re measure of a convex function u implies the function must actually be strictly convex. A lesser known result, also by Caffarelli, states that if the Monge-Amp{\`e}re of u has only a lower bound, the contact set between u and a supporting affine function must have affine dimension strictly less than n/2. By making a geometric construction involving the subdifferential of a convex function at a singular point, we give an alternative proof of Caffarelli's result. Additionally, this method can be used to extend the result to optimal transport problems with cost functions satisfying the weak Ma-Trudinger-Wang condition. This talk is based on a joint work with Young-Heon Kim.

##### Minimal and constant mean curvature surfaces in quotients of H^2xR

In this talk we will report on two types of questions related to the geometry of surfaces in quotients of H^2xR. We will talk about the classification of compact embedded constant mean curvature surfaces, and about the description of the asymptotic behavior of ends of proper minimal surfaces with finite total curvature. These theorems are inspired by classical results in Euclidean space.

##### Long time existence of minimizing movement solutions of Calabi flow

In 1982 Calabi proposed studying gradient flow of the L^2 norm of the scalar curvature (now called Calabi flow) as a tool for finding canonical metrics within a given Kahler class. The main motivating conjecture behind this flow (due to Calabi-Chen) asserts the smooth long time existence of this flow with arbitrary initial data. By exploiting aspects of the Mabuchi-Semmes-Donaldson metric on the space of Kahler metrics I will construct a kind of weak solution to this flow, known as a minimizing movement, which exists for all time.

##### A resolution of the Yang-Mills-Dirichlet Problem in super-critical dimensions'

In the early 2000 a series of geometric works, by Donaldson-Thomas and Tian in particular, have stimulated the analysis of Yang-Mills fields in dimension larger than the conformal one. We shall recall the main ingredients, developed mostly in the 80's, for the study of Yang-Mills Fields in critical and sub-critical dimensions. PLEASE CLICK ON SUMMARY TITLE FOR COMPLETE ABSTRACT.

##### Regularity theory for area-minimizing currents

It was established by Almgren at the beginning of the eighties that area-minimizing $n$-dimensional currents in Riemannian manifolds are regular up to a singular set of dimension at most $n-2$. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Embedded constant mean curvature surfaces in Euclidean three space

Constant mean curvature (CMC) surfaces are critical points to the area functional with an enclosed volume constraint. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### On equations with drift and diffusion.

An drift-diffusion equation is like the heat equation with an extra first order term. In some cases, the Laplacian is replaced by a fractional Laplacian. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Long-time analysis of 3 dimensional Ricci flow

It is still an open problem how Perelman's Ricci flow with surgeries behaves for large times. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Compactification, projective geometry, and Einstein metrics

Conformal compactification, as originally defined by Penrose, has long been recognised as an effective geometric framework for relating conformal geometry, and associated field theories ``at infinity'', to

the asymptotic phenomena of an interior (pseudo-)-Riemannian geometry of one higher dimension. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Isoperimetric inequality and Q-curvature

Using the techniques of $A_p$ weights, we study the relationships between the isoperimetric inequality with the Paneitz Q-curvature. We show a Fiala-Huber type isoperimetric inequality for higher dimensions in which

the isoperimetric constant depends only on the integrals of the Q-curvature.

##### From Global to Local

In this talk we discuss aspects of the recent proof of the Caratheodory conjecture on the number of umbilic points on a closed convex surface in Euclidean 3-space. The proof starts with the global version of the conjecture for closed surfaces and ultimately leads to an index bound for the principal foliation of the surface about isolated umbilic points. This reverses the historic perspective on the conjecture and opens up a dichotomy between smooth and real analytic surfaces. The methods used include mean curvature flow with boundary, holomorphic discs and utilizes a Kaehler structure in which the metric is of indefinite signature (2,2).

##### Rigidity of Self-shrinkers Asymptotic to Cylinders

In this talk, we show the self-shrinkering end that are asymptotic to the shrinking cylinders in a certain sense must be isometric to the cylinders. Also, we show the asymptotic condition is optimal. Our result holds for all dimensions.

##### On Liouville type theorems

**PLEASE NOTE SPECIAL TIME. **We will review the general theme of regularity theory and the importance of global solutions In the whole theory. Particularly we will study the situation in a cylinder and upper half space for uniformly elliptic equations either in divergent form or nondivergent form. We show, amount other things the any positive solution has to exponential grow in at least one direction and certain uniqueness.** **