# Seminars & Events for Differential Geometry & Geometric Analysis Seminar

##### Coarse Ricci curvature and the manifold learning problem

We use the framework exploited by Bakry and Emery for the study of logarithmic Sobolev inequalities to consider the statistical problem of estimating the Ricci curvature (and more generally the Bakry-Emery tensor of a manifold with density) of an embedded submanifold of Euclidean space from a point cloud drawn from the submanifold. Our method leads to a definition of Coarse Ricci curvature alternative to the one proposed by Y. Ollivier. This is joint work with Micah Warren.

##### On the topology and index of minimal surfaces

We show that for an immersed two-sided minimal surface in R^3, there is a lower bound on the index depending on the genus and number of ends. Using this, we show the nonexistence of an embedded minimal surface

in R^3 of index 2, as conjectured by Choe. Moreover, we show that the index of an immersed two-sided minimal surface with embedded ends is bounded from above and below by a linear function of the total curvature of the surface. (This is joint work with Otis Chodosh)

##### Umbilicity and characterization of Pansu spheres in the Heisenberg group

For n≥2 we define a notion of umbilicity for hypersurfaces in the Heisenberg group H_{n}. We classify umbilic hypersurfaces in some cases, and prove that Pansu spheres are the only umbilic spheres with positive constant p(or horizontal)-mean curvature in H_{n} up to Heisenberg translations. This is joint work with Hung-Lin Chiu, Jenn-Fang Hwang, and Paul Yang.

##### Regularity of manifolds with bounded Ricci curvature and the codimension $4$ conjecture

This talk will concern joint work with Aaron Naber on the regularity of noncollapsed Riemannian manifolds $M^n$ with bounded Ricci curvature $|{\rm Ric}_{M^n}|\leq n-1$, as well as their Gromov-Hausdorff limit spaces, $(M^n_j,d_j)\stackrel{d_{GH}}{\longrightarrow}(X,d)$, where $d_j$ denotes the Riemannian distance. We will explain a proof of the conjecture that $X$ is smooth away from a closed subset of codimension $4$. By combining this with the ideas of quantitative stratification, we obtain a priori $L^q$ estimates on the full curvature tensor, for all $q<2$. We also prove a conjecture of Anderson stating that for all $v>0$, $ <\infty$, the collection of $4$-manifolds $(M^4,g)$ with $|{\rm Ric}_{M^n}|\leq 3$, ${\rm Vol}(M^3)\geq v$, ${\rm diam}(M^4)\leq d$, contains a most a finite number of diffeomorphism types.

##### Nonexistence results for solitons in the mean curvature flow

In this talk I will give a more refined quantitative understanding of some of the important known solitons in the n-dimensional mean curvature flow in R^{n+1}. The global estimates in question follow by iteration of monotonicity formulae - an idea and technique which, while quite elementary in nature, appears to be particularly useful in several of such situations. The applications of the explicit estimates are many. I will focus mostly on the new nonexistence theorems that follow. Time permitting, I will also mention other, related consequences for the analysis of the soliton PDEs for the flow.

##### Regularity of semi-calibrateed integral $2$-cycles

Semi-calibrated currents naturally appear when dealing with several geometric questions, some aspects of which require a deep understanding of regularity properties of semi-calibrated currents. We will focus on the case of dimension $2$, where it turns out that semi-calibrated cycles are actually pseudo holomorphic. By using an analysis implementation of the algebro-geometric blowing up of a point we study the regularity of semi-calibrated $2$-cycles from the point of view of uniqueness of tangent cones and of local smoothness.

##### Shear-free asymptotically hyperboloidal initial data in general relativity

One of the most useful tools for studying isolated gravitational systems is conformal compactification, which transforms ``infinity'' into a finite boundary by multiplying the metric by a suitable smooth function. For studying outgoing radiation effects, an effective approach is to set up an initial-value problem on an asymptotically hyperboloidal spacelike hypersurface that intersects future conformal null infinity transversely. The initial data for the Einstein equations consists of a Riemannian metric and a second fundamental form on the initial hypersurface, together with some matter and energy fields. These data are not freely specifiable, but instead must satisfy the Einstein constraint equations, which boil down to a nonlinear elliptic system on the chosen initial hypersurface.

##### Asymptotic geometry and large isoperimetric regions

**This is an additional talk on this date. **In spite of the long history of the isoperimetric problem, the list of manifolds where isoperimetric regions are well understood is remarkably short. In this talk, I will discuss the following question: "If a Riemannian manifold (M,g) is asymptotic at infinity to a space in which the isoperimetric regions are understood, can we determine the large isoperimetric regions in (M,g)?" I'll discuss some positive and negative answers to this question for various asymptotic geometries. Some of these results are joint work with Michael Eichmair and Alex Volkmann.

##### Quantitative uniqueness, doubling lemma and nodal sets

Based on a variant of frequency function, we improve the vanishing order of solutions for Schr\"{o}dinger equations which describes quantitative behavior of strong uniqueness continuation property. For the first time, we investigate the quantitative uniqueness of higher order elliptic equations and show the vanishing order of solutions. Using the Carleman estimates, we obtain the doubling estimates and optimal vanishing order of Steklov eigenfunctions, which is the eigenfunctions of the Dirichlet-to-Neumann map. A lower bound of nodal sets of Steklov eigenfunctions is also derived.

##### Pluriclosed flow and generalized Kahler geometry

In joint work with G. Tian I introduced a natural evolution equation generalizing the Kahler Ricci flow to complex, non-Kahler manifolds. Moreover we showed that this equation preserves "generalized Kahler geometry." In this talk I will discuss further results on this flow in the generalized Kahler setting, including a sharp long time existence result for complex surfaces. These results lead to strong rigidity and classification results for generalized Kahler structures.

##### Sign of Green's function of Paneitz operator and the Q curvature equations

I will review some recent works with Paul on the fourth order Paneitz operators and Q curvature equations. Among other things, we will discuss the positivity of Green's function of Paneitz operator and its applications to finding constant Q curvature metrics by variational methods.

##### Global existence for fully nonlinear stochastic hyperbolic equations : the Nash-Moser approach

In 1979, Klainerman proved the first global existence result for fully nonlinear hyperbolic equations using Nash-Moser's method. The result was a breakthrough in the field.

In the present talk, I shall report on the corresponding study for fully nonlinear stochastic hyperbolic equations driven by a Wiener process. The main result of the research is stochastic counterpart of Klainerman's global existence theorem. The main tool for our investigation is Nash-Moser's iteration scheme combined with stochastic calculus and the theory of enlargement of filtrations.

##### Minimal surfaces in H2xR

**PLEASE NOTE SPECIAL DAY AND LOCATION: Talk #1. **In the classical theory of minimal surfaces of the Euclidean space, the better known ones are those with finite total curvature. Laurent Hauswirth and Harold Rosenberg started in 2006 the corresponding theory of complete minimal surfaces with finite total curvature in H2xR, the product space of the hyperbolic plane and the real line. We will give an overview of the subject, including the construction of some examples and

classification results.

##### A positive mass theorem for asymptotically flat manifolds with a noncompact boundary and an application to a Yamabe-type flow

**PLEASE NOTE SPECIAL DAY AND LOCATION: Talk #2. **First I will discuss a positive mass theorem for noncompact manifolds with boundary and ends asymptotic to the Euclidean half-space. This theorem is a joint work with Ezequiel Barbosa and Levi de Lima.Â Then I will show how this result can be applied to prove the convergence of aÂ certain Yamabe-type flow on compact manifolds with boundary.** **

##### Mean curvature flow without singularities

**Please note special day, time and location. **We study graphical mean curvature flow of complete solutions defined on subsets of Euclidean space. We obtain smooth long time existence. The projections of the evolving graphs also solve mean curvature flow. Hence this approach allows to smoothly flow through singularities by studying graphical mean curvature flow in one additional dimension.

##### Kähler-Einstein metrics: from cones to cusps

In this talk, I will explain the following result and outline its proof: Let $X$ be a compact Kähler manifold and $D$

a smooth divisor such that $K_X+D$ is ample. Then the negatively curved Kähler-Einstein metric with cone angle $\beta$ along $D$ converges to the cuspidal Kähler-Einstein metric of Tian-Yau when $\beta$ tends to zero.

##### Duality in Boltzmann Equation and its Applications.

In this paper we will survey a quantitative and qualitative

development on the Boltzmann equation. This development reveals the dual

natures of the Boltzmann

equation: The particlelike nature and the fluidlike nature. This dual

nature property gives rise to the precise construction of the Green's

function for Boltzmann equation around a global Maxwellian state. With the

precise structure of the Green's function, one can implement the Green's

function to study various problems such as invariant manifolds for the

steady Boltzmann flows, time asymptotic nonlinear stability of Boltzmann

shock layers and Boltzmann boundary layers, Riemann Problem, and

bifurcation problem of boundary layer problem, etc.

##### Gromov-Haudorff convergence of Kahler manifolds and the finite generation conjecture

We study the uniformization conjecture of Yau by using the Gromov-Haudorff convergence. As a consequence, we confirm Yau's finite generation conjecture. More precisely, on a complete noncompact Kahler manifold ith nonnegative bisectional curvature, the ring of polynomial growth holomorphic functions is finitely generated. During the course, we prove if M is a complete noncompact Kahler manifold with nonnegative bisectional curvature and maximal volume growth, then it is biholomorphic to an affine algebraic variety. We also confirm a conjecture of Ni on the equivalence of several conditions on complete Kahler manifolds with nonnegative bisectional curvature.

##### Minimal Surfaces with Arbitrary Topology in H^2xR

In this talk, we show that any open orientable surface can be embedded in H^2xR as a complete area minimizing surface. Furthermore, we will discuss the asymptotic Plateau problem in H^2xR, and give a fairly complete solution.

##### Levi-flat hypersurfaces in complex manifolds

Levi-flat hypersurfaces arise naturally from complex foliation theory and complex dynamics. In this talk we will discuss the Cauchy-Riemann equations on domains in complex manifolds with Levi-flat boundary. We give an example of a pseudoconvex domain in a complex manifold whose $L^2$-Dolbeault cohomology is non-Hausdorff, yet the domain is Stein with real-analytic Levi-flat boundary (Joint work with Debraj Chakrabarti).