# Seminars & Events for Differential Geometry & Geometric Analysis Seminar

##### Cornered Asymptotically Hyperbolic Metrics

This talk will concern cornered asymptotically hyperbolic spaces, which have a finite boundary in addition to the usual infinite boundary. After introducing the setting, we will present a normal form near the corner for these spaces. Using this, we will then discuss formal existence and uniqueness, near the corner, of asymptotically hyperbolic Einstein metrics, with a CMC-umbilic condition imposed on the finite boundary.

##### The degenerate special Lagrangian equation

I will discuss a degenerate form of the special Lagrangian equation that arises as the geodesic equation for the space of positive Lagrangians. Considering graph Lagrangians in Euclidean space, the equation reduces to a second order fully non-linear PDE for a single real function. I will explain how to prove existence and uniqueness for Lipschitz solutions to the Dirichlet problem on a convex domain times the unit interval. The proof uses the subequation theory of Harvey-Lawson. Existence of solutions on general manifolds with sufficient regularity would imply a version of the strong Arnold conjecture in Hamiltonian dynamics as well as uniqueness for special Lagrangians. This talk is based on joint work with Yanir Rubinstein.

##### Fill-ins, extensions, scalar curvature, and quasilocal mass

There is a special relationship between the Jacobi operator and the ambient scalar curvature operator, which we'll exploit. First, I'll talk about a "cut-and-fill" technique that simplifies 3-manifolds of nonnegative scalar curvature. This was used in the study of a priori L^1 estimates for the boundary mean curvature of mean-convex fill-ins with nonnegative scalar curvature, and to generalize Brown-York mass. Second, I'll talk about an extremal bending technique that lets us compute the Bartnik mass of apparent horizons. Parts of this talk reflect work done jointly with R. Schoen/P. Miao.

##### Generated Jacobian equations and regularity: optimal transport, geometric optics, and beyond

**PLEASE NOTE SPECIAL DAY, TIME AND LOCATION. **Equations of Monge-Amp{\`e}re type arise in numerous contexts, and solutions often exhibit very subtle properties; due to the highly nonlinear nature of the equation, and its degenerate ellipticity. Motivated by an example from geometric optics I will talk about the class of Generated Jacobian Equations, recently introduced by Trudinger. This class includes optimal transport, the Minkowski problem, and the classical Monge-Amp{\`e}re equation. I will present a new regularity result for weak solutions of these equations, which is new even in the case of equations arising from near-field problems in geometric optics. This talk is based on joint works with Nestor Guillen.

##### The Cauchy-Riemann equations in complex manifolds

In this talk we will discuss the Cauchy-Riemann equations on domains in complex manifolds with positive or negative curvature. We will also report some recent new results on the $L^2$ closed range property for $\bar{\partial}$ on an annulus between two pseudoconvex domains, when the inner domain is not smooth. In particular, we show the Hausdor property of the $L^2$ Dolbeault cohomology group on a domain between a ball and a bi-disc, the so-called Chinese Coin problem. We also give characterizaation of Lipschitz domains with holes through their Dolbeault cohomology groups. (joint work with Debraj Chakrabarti, Siqi Fu, and Christine Laurent-Thiebaut).

##### The free-boundary Brakke flow

A surface has geometric free-boundary if it meets some barrier hypersurface orthogonally, like a bubble on a bathtub. We extend Brakke's weak notion of mean curvature flow to have a free-boundary condition, which allows the surface to ``pop'' upon tangential contact with the barrier.

##### What We Know and Don’t Know about the Space of Solutions of the Einstein Constraint Equations

Ten years ago, Robert Bartnick and I wrote a review article summarizing what was known at the time about the Einstein constraint equations and their solutions. In that article, we noted that while much was known about solutions of the constraints which have constant mean curvature (CMC) or are nearly CMC, very little was known about solutions which are far from CMC. The hope at the time was that the effectiveness of the conformal method for constructing CMC and near-CMC solutions would (perhaps after much work) extend to far-from-CMC solutions. In the years since that article appeared, new results have slowly been obtained. While some have been consistent with this optimistic view, many others have shown that the picture for far-from CMC solutions is likely to be much more complicated.

##### Kahler-Einstein metrics and volume minimization

Futaki invariant is an obstruction to the existence of Kahler-Einstein metrics on Fano manifolds. Martelli-Sparks-Yau showed that the Futaki invariant is the derivative of a normalized volume functional on the space of Reeb vector fields of associated affine cones and derived the volume minimization principle for more general Sasaki-Einstein metrics. I will show that this volume minimization principle can be extended to work on a much bigger space of centered real valuations. This gives an equivalent characterization of K-semistability (which is equivalent to ``almost Kahler-Einstein”) and has an interesting algebra-geometric consequence. If time permits, I will also discuss the generalization to the case of Sasaki-Einstein metrics and some relation to metric tangent cones on singular Kahler-Einstein varieties.

##### Proof of a Null Penrose Conjecture using a new Quasi-local Mass

We define an explicit quasi-local mass functional which is nondecreasing along all null foliations (satisfying a convexity assumption) of null cones. We use this new functional to prove the Null Penrose Conjecture under fairly generic conditions.

##### Uniqueness of weak solutions to the Ricci flow

In his resolution of the Poincaré and Geometrization Conjectures, Perelman constructed Ricci flows in which singularities are removed by a surgery process. His construction depended on various auxiliary parameters, such as the scale at which surgeries are performed. At the same time, Perelman conjectured that there must be a canonical flow that automatically "flows through its surgeries”, at an infinitesimal scale. Recently, Kleiner and Lott constructed so-called Ricci flow space-times, which exhibit this desired behavior. In this talk, I will first review their construction. I will then present recent work of Bruce Kleiner and myself, in which we show that these Ricci flow space-times are in fact unique and fully determined by their initial data.

##### Properly immersed CMC surfaces in hyperbolic 3-manifolds of finite volume

**Please note special time: 2:00.** If $N$ is a noncompact hyperbolic 3-manifold of finite volume and $\Sigma$ is a properly immersed surface of finite topology with nonnegative constant mean curvature less than 1, then we prove that each end of $\Sigma$ is asymptotic (with finite positive multiplicity) to a totally umbilic annulus, properly embedded in $N$.

##### A Minkowski inequality for hypersurfaces in the Anti-deSitter-Schwarzschild manifold

We prove a sharp inequality for hypersurfaces in the Anti-deSitter-Schwarzschild manifold. This inequality generalizes the classical Minkowski inequality for surfaces in the three dimensional Euclidean space, and has a natural interpretation in terms of the Penrose inequality for collapsing null shells of dust. The proof relies on a monotonicity formula for inverse mean curvature flow, and uses a geometric inequality established by Brendle.

##### SPECIAL LECTURE: Results Related to the CR Yamabe Problem

**This is a special lecture in Geometric Analysis. Please note different day, time and location.** I will talk about the CR Yamabe problem and its related results, including the CR Yamabe flow, uniqueness and compactness of the CR Yamabe problem. I will also talk about its various generalizations, including the problem of prescribing the Webster scalar curvature and the CR Yamabe problem on noncompact manifolds.

##### Optimal regularity of three dimensional mass minimizing cones

In this talk I will present the following regularity results in minimal surface theory: the singular points of a mass minimizing three dimensional cone in the Euclidean space are contained in at most finitely many half lines (joint with C. De Lellis and L. Spolaor).

##### A Feynman-Kac formula for differential forms on manifolds with boundary and applications

We prove a Feynman-Kac-type formula for the heat flow acting on differential forms satisfying absolute boundary conditions on Riemannian manifolds with boundary and of bounded geometry. We use this to construct L^2 harmonic forms out of bounded ones on the universal cover of a compact Riemannian manifold whose geometry displays a positivity property expressed in terms of a certain stochastic average of the Weitzenbock operator acting on forms and the second fundamental form of the boundary. As a geometric application we find a new obstruction to the existence of metrics with positive isotropic curvature and 2-convex boundary.We also present a version of the Feynman-Kac formula for spinors under suitable boundary conditions and discuss potential applications.

##### Asymptotic structure of self-shrinkers

Self-shrinkers are singularity models for mean curvature flow. In this talk, I will show that each end of a noncompact self-shrinker in the Euclidean three-space of finite topology is smoothly asymptotic to a regular cone or a round cylinder.

##### Conformal metrics with constant scalar curvature and constant boundary mean curvature

**This is a special Geometric Analysis seminar. Please note special day, time and location. **Analogous to the Yamabe problem, a very natural question on a compact manifold with boundary is deforming Riemannian metrics to conformal ones with constant scalar curvature and constant mean curvature curvature on the boundary. Escobar proved the existence of such conformal metric when the dimension is 3,4 and 5 or the boundary is not umbilic or the Weyl tensor does not vanish on the boundary. We generalized his result to the case when dimension is 6 and 7. Some other remaining cases left open by Escobar are also considered. I will also introduce the Han-Li conjecture related to this problem. I will show that Han-Li conjecture is true under some conditions. This is a joint work with Professor Xuezhang .

##### The isoperimetric problem for Lens spaces

Given a Riemannian manifold, the isoperimetric problem consists in classifying the regions that minimize perimeter among regions of same volume. In this talk, we show that the solutions of the isoperimetric problem in the Lens space with large fundamental group are either geodesic spheres or tori of revolution about geodesics.

##### On a fully nonlinear version of the Min-Oo Conjecture

**Please note: This is an additional DGGA seminar for this date. **In this talk, we show rigidity results for super-solutions to fully nonlinear elliptic conformally invariant equations in subdomains of the standard $n$-sphere $\s^n$ under suitable conditions on the boundary. This proves rigidity for compact connected locally conformally flat manifolds $(M,g)$ with boundary such that the eigenvalues of the Schouten tensor satisfy a fully nonlinear elliptic inequality and whose boundary is isometric to a geodesic sphere $\partial D(r)$, $D(r)$ a geodesic ball of radius $r\in (0,\pi/2]$ in $\mathbb{S}^n$, and totally umbilic with mean curvature bounded bellow by the mean curvature of this geodesic sphere. Under the above conditions, $(M,g)$ must be isometric to the closed geodesic ball $\overline{D(r)}$.

##### Appearance of stable spheres along the Ricci flow in positive scalar curvature

For a 3-manifold M not isometric to the round sphere, with scalar curvature at least 6 and positive Ricci curvature, Marques and Neves proved a 3-dimensional version of the Toponogov theorem: there exists an embedded minimal surface S of area less than 4pi. Their proof uses a combination of min-max theory for minimal surfaces and the Ricci flow. While the general case (no assumption on the Ricci curvature) can now be proved with a rather different approach, it is desirable to extend the Ricci flow method, for it yields more geometric information on S. When trying to do so, a natural question arises. Suppose that the scalar curvature is positive at time 0, can stable surfaces appear along the Ricci flow if there were none of them at time 0? I will show examples where not only stable spheres appear but a non-trivial singularity occurs.