# Seminars & Events for Differential Geometry & Geometric Analysis Seminar

##### Singular critical points for variational free boundary problem from isoparametric hypersurfaces

Consider the following variational free boundary problem: \min J(u)=\int_B (|\nabla u|^2+\chi_{\{u>0\}})dx, where $B$ is the unit ball, and the boundary value of $u$ on $\partial B$ is prescribed. The main problem is the regularity for the free boundary $F(u)=\partial\{u>0\}$ for a minimizer $u$. $J$ has a trivial minimizer $u(x)=x_n, x_n>0$, $u(x)=0,x_n\leq 0$. As in the theory of minimal surface, regularity of free boundary is equivalent to existence of singular minimal cone, that is, nontrivial minimizer of $J$ of degree 1: $u(rx)=ru(x)$. We construct three families of singular critical points for $J$, which are homogeneous of degree 1. The level sets of these solutions on the unit sphere are isoparametric surfaces and their focal submanifolds.

##### On a remarkable formula of Jerison and Lee in CR geometry

I will discuss a remarkable formula discovered by Jerison and Lee to classify constant scalar curvature pseudohermitian structures on the sphere. We show that the formula is valid in the wider context of Einstein pseudohermitian manifolds. As an application we prove a uniqueness result that generalizes the theorem of Jerison and Lee.

##### A compactness result for Fano manifolds and K\"ahler Ricci flows

**Please note different day (Thursday). **In this talk, we present a joint result with G.Tian. It is a convergence-compactness result for Fano manifolds and K\"ahler Ricci flows. Comparing to the more general Riemannian versions, in the Fano case, our curvature assumption is much weaker and is preserved by the K\"ahler Ricci flows. As one application, we obtain a new local regularity criteria and structure result for K\"ahler Ricci flows.** **

##### Stationary Kirchhoff systems in closed manifolds

We discuss various issues concerning stationary Kirchhoff systems in closed manifolds such as existence, nonexistence, and compactness. The nonlocal nature and the vector valued aspect of the equations lead to surprising results.

##### Uniqueness of blowups and Lojasiewicz inequalities

Once one knows that singularities occur, one naturally wonders what the singularities are like. For minimal varieties the first answer, already known to Federer-Fleming in 1959, is that they weakly resemble cones. For mean curvature flow, by the combined work of Huisken, Ilmanen, and White, singularities weakly resemble shrinkers. Unfortunately, the simple proofs leave open the possibility that a minimal variety or a mean curvature flow looked at under a microscope will resemble one blowup, but under higher magnification, it might (as far as anyone knows) resemble a completely different blowup.

##### Interior curvature estimates and the asymptotic plateau problem

**Please note special day (Thursday), time and location. **In this talk, we will begin with the definition of hyperbolic space. After giving some known results for constant curvature hypersurfaces in hyperbolic space, we will state our main result and also sketch how it's proven. If we have time, we will show a strong duality theorem of hyperbolic space and De Sitter space. We will also show some applications of this duality theorem. This work is joint with Bo Guan and Joel Spruck.** **

##### Curvature estimates for immersed submanifolds in Space form

**Please note special day (Tuesday), time and location. **The regularity of immersed submanifolds depends on the estimates on the extrinsic curvatures. In this talk, we discuss some new estimates for the second fundamental forms of embedded submanifolds in terms of intrinsic geometry. Some of these results are even new for the hypersurfaces (co-dimension one case). Key ingredients are drawn from recent works in the problem of prescribed curvature measures and regularity of curvature equations.

##### Berger decomposition, Weitzenbock formula and canonical metrics on four-manifolds

In this talk we will first provide an alternative proof of the Weitzenbock formula using Berger curvature decomposition, as applications we will discuss rigidity of Einstein four-manifolds under some positive curvature

conditions. At the end we will discuss the Weitzenbock formula for generalized quasi-Einstein metrics on four-manifolds.

##### Bounds on the Bondi mass in perturbations of the Schwarzschild exteriors

**Please note special time (4:15). **We consider vacuum perturbations of the Schwarzschild black hole exteriors and derive bounds on the Bondi mass and energy at past null infinity in terms of the area of certain carefully chosen sections of the apparent horizon. The method relies on a perturbation argument; in particular it only requires a slab of space-time around a smooth null surface originating on the apparent horizon and terminating at past null infinity.** **

##### Structure of measures in Lipschitz differentiability spaces

This talk will present results showing the equivalence of two very different ways of generalising Rademacher's theorem to metric measure spaces. The first was introduced by Cheeger and is based upon

differentiation with respect to another, fixed, chart function. The second approach is new for this generality and originates in some ideas of Alberti. It is based upon forming partial derivatives along a very

rich structure of Lipschitz curves, analogous to the differentiability theory of Euclidean spaces. By examining this structure further, we naturally arrive to several descriptions of Lipschitz differentiability

spaces.

##### Asymptotics of minimal submanifolds in AdS/CFT correspondence

The talk will describe the asymptotics of minimal submanifolds in spaces which are the product of an asymptotically hyperbolic manifold and a compact Riemannian manifold. Such spaces arise in the AdS/CFT correspondence in physics. Results include the derivation of a minimality constraint on the boundary submanifold and an identification of the local data necessary to determine the asymptotics. Examples will be presented. This is joint work with Andreas Karch.