# Seminars & Events for Differential Geometry & Geometric Analysis Seminar

##### CMC foliations on alomost Fuchsian 3-manifolds

An alomost Fuchsian 3-manifold is a quasi-Fuchsian 3-manifold that contains an incompressible surface with principal curvatures in (-1,1). In this talk, we will show that any alomost Fuchsian 3-manifold admits a CMC foliation.

##### A parabolic flow of Hermitian metrics

I will introduce a parabolic flow of Hermitian metrics which is a generalization of Kähler-Ricci flow. This flow preserves the pluriclosed condition, and its existence and convergence properties are closely related to the underlying topology of the given complex manifold. I will discuss a stability result for the flow near Kähler-Einstein metrics. Further, I will classify static solutions to the flow on various classes of complex surfaces, and show that no static solutions exist on Class VII surfaces. Finally I will discuss possible applications of this flow to understanding the topology of nonKahler surfaces. Joint with G. Tian.

##### Hamiltonian Stationary Tori in Kahler Manifolds

A solution of the Hamitonian stationary variational problem is a Lagrangian submanifold (of a symplectic manifold with compatible Riemannian metric) whose volume is stationary under Hamiltonian variations. In joint work with Adrian Butscher, we construct families of small Hamiltonian stationary Lagrangian tori near a point in a Kahler four-manifold, under a non-degeneracy condition on the complex curvature tensor at the point.

##### Pseudo-Riemannian Calibrated Geometry and Optimal Transportation

Given a manifold $M$, there is a naturally occurring pseudo-Riemannian metric and Kähler form on the product $M\times M$. The graph of the solution to the optimal transportation problem for given smooth densities on $M$ is then a calibrated maximal Lagrangian submanifold in $M\times M$, with respect to a conformal metric on $M\times M$. Thus the graph of the optimal map is special Lagrangian in the sense of Hitchin. This variational characterization of optimal transportation is different from the traditional one. The calibrations which detect these special Lagrangians are pseudo-Riemannian analogues of the special Lagrangian calibrations for Calabi-Yau manifolds. Like in the Calabi-Yau case, the moduli space of such submanifolds is itself a manifold of dimension $b_1(M)$.

##### The Static Extension Problem in General Relativity

There are several competing definitions of quasi-local mass in General Relativity. A very promising and natural candidate, proposed by Bartnik, seeks to localize the ADM or total mass. Fundamental to understanding Bartnik's construction, is the question of existence for a canonical geometric boundary value problem associated with the static vacuum Einstein equations. In this talk we report on joint work with M. Anderson, which answers this question affirmatively under the hypothesis of a certain nondegeneracy condition. We also show that uniqueness fails.

##### Singularities, test configurations and constant scalar curvature Kahler metrics

##### The Yamabe problem on manifolds with boundary

##### Center of mass and constant mean curvature foliations in general relativity

We will discuss the existence and uniqueness of the foliation by stable spheres with constant mean curvature for asymptotically flat manifolds satisfying the Regge-Teitelboim condition at infinity. We will first introduce the background and then discuss how the foliation relates to the concept of center of mass in general relativity.

##### The geometry of static and stationary matter configurations in relativity

There are a number of interesting questions and some recent results concerning the geometry of equilibrium matter and black hole configurations for the Einstein equations. In this talk we will give background on these questions and survey known results. We will then describe recent work with R. Beig and G. Gibbons on the problem.

##### Surface Comparison With Mass Transportation

A method for transportation of metric between simply-connected surfaces with boundary is presented. The method is based on classical uniformization and optimal mass transportation. One application of the method is a novel definition of distance function between simply-connected surfaces with boundaries. The new distance forms a tractable/constructive alternative to the theoretically sound but hard to compute Gromov-Hausdorff distance. We will also show how this distance can be used to find automatic correspondences between some "real-life" surfaces.

##### Conformal Structure of Minimal Surfaces with Finite Topology

The recent construction of a genus-one helicoid verified the existence of a second example of a complete, embedded minimal surface with finite topology and infinite total curvature in $\mathbb{R}3$. We determine the conformal structure and asymptotic Weierstrass data of all surfaces with these properties. Using this structure and the asymptotics, in the case $g=1$ we establish the existence of an orientation preserving isometry. This is joint work with Jacob Bernstein

##### Uniqueness of constant scalar curvature Kähler metrics

We will show that constant scalar curvature Kähler(cscK) metric "adjacent" to a given Kähler class is unique up to isomorphism. This generalizes the previous uniqueness theorems of Chen-Tian and Donaldson, where the complex structure is fixed. Joint work with X-X. Chen.

##### On m-Quasi Einstein metrics

We say an $n$-dimensional Riemannian manifold is an $m$-Quasi Einstein metric if it is the base of an $(n+m)$-dimensional warped product Einstein manifold. We view the $m$-Quasi Einstein equation as a generalization of the Einstein equation (since an Einstein manifold is the base of a trivial product Einstein manifold). The $m$-Quasi Einstein equation is also closely related to the gradient Ricci soliton equation. In this talk I will give an overview of some earlier results about the classification of $m$-quasi Einstein metrics and prove a new classification of $m$-Quasi Einstein metrics with harmonic curvature. This is joint work with Peter Petersen and Chenxu He.

##### Optimal conditions for the extension of the mean curvature flow

In this talk, we will discuss several optimal (global) conditions for the existence of a smooth solution to the mean curvature flow. Our focus will be on quantities involving only the mean curvature. We will also discuss several applications of a local curvature estimate which is a parabolic analogue of Choi-Schoen estimate for minimal submanifolds. This is joint work with Natasa Sesum.

##### Complete Calabi-Yau metrics from rational elliptic surfaces

A rational elliptic surface is the blow-up of P2 in the nine base points of a pencil of cubics. The pencil then lifts as a fibration of the surface by elliptic curves. I show that the complement of any fiber F admits families of complete Calabi-Yau metrics, whose asymptotic geometry depends in a delicate way on the monodromy of the fibration around F. If F is smooth, these metrics all converge to flat cylinders at an exponential rate, and in that case I give a complete description of the local Einstein deformation space.

##### Conformally Warped Manifolds and quasi-Einstein metrics

The concept of a smooth metric measure space has recently arisen as a useful object within Riemannian geometry, for example in Perelman's formulation of Ricci flow as a gradient flow. In this setting, a key objective is to find a suitable generalization of Ricci curvature, and to understand the associated ``quasi-Einstein'' metrics. Taking two different perspectives, Lott, Villani, Sturm and Chang, Gursky and Yang have found two distinct approaches to studying smooth metric measure spaces. While the formulations are different, they both introduce an extra dimensional parameter $m$ which, in the limit $m\to\infty$, recovers the curvatures that arise in Perelman's treatment of the Ricci flow. In this way it becomes interesting to see if the two approaches are related.

##### The new Intrinsic flat distance between oriented Riemannian manifolds

We define a new distance between oriented Riemannian manifolds that we call the "intrinsic flat distance" based upon Ambrosio-Kirchheim's theory of integral currents on metric spaces. Limits of sequence of manifolds with a uniform upper bound on their volume and diameter are countably H^m rectifiable metric spaces with an orientation and multiplicity that we call "integral current spaces". In general the Gromov-Hausdorff and intrinsic flat limits do not agree. Intrinsic flat convergence is a weaker notion. We show that they do agree when the sequence of manifolds has nonnegative Ricci curvature and a uniform lower bound on volume and also when the sequence of manifolds has a uniform linear local geometric contractibility function.

##### A Codazzi-like equation and the singular set for surfaces in the Heisenberg group

##### p-harmonic forms on complete manifolds

Let $M$ be an m-dimensional complete non-compact Reimannian manifold. We prove that any bounded set of p-harmonic k-forms in $L^q(M)$, is relatively compact with respect to the uniform convergence topology if the curvature operator of $M$ is asymptotically non-negative.

##### Ricci flow and the determinant of the Laplacian on non-compact surfaces

The determinant of the Laplacian is an important invariant of closed surfaces and has connections to the dynamics of geodesics, Ricci flow, and physics. Its definition is somewhat intricate as the Laplacian has infinitely many eigenvalues. I'll explain how to extend the determinant of the Laplacian to non-compact surfaces where one has to deal with additional difficulties like continuous spectrum and divergence of the trace of the heat kernel. On surfaces (even non-compact) this determinant has a simple variation when the metric varies conformally. I'll explain how to use Ricci flow to see that the largest value of the determinant occurs at constant curvature metrics.This is joint work with Clara Aldana and Frederic Rochon.