# Seminars & Events for Differential Geometry & Geometric Analysis Seminar

##### Harmonic Chern Forms on Polarized Kähler Manifolds

The higher K-energies are functionals whose critical points give Kähler metrics with harmonic Chern forms. In this talk, we relate the higher K-energies to discriminants and use the theory of stable pairs to obtain results on their boundedness and asymptotics.

##### The Lojasiewicz-Simon gradient inequality and its applications to discreteness of the energy spectrum for harmonic maps

The Lojasiewicz-Simon gradient inequality is a generalization, due to Leon Simon (1983), to analytic or Morse-Bott functionals on Banach manifolds of the finite-dimensional gradient inequality, due to Stanislaw Lojasiewicz (1963), for analytic functions on Euclidean space. In this talk, we shall discuss several generalizations of the Lojasiewicz-Simon gradient inequality and a selection of their applications, including discreteness of the energy spectrum for harmonic maps from Riemann surfaces into analytic Riemannian manifolds. This is joint work with Manos Maridaks.

##### Mean convex level set flow in general ambient manifolds: MOVED TO MARCH 11

**PLEASE NOTE: THIS SEMINAR HAS BEEN MOVED TO MARCH 11, 2016.**

##### The hypoelliptic Laplacian

The hypoelliptic Laplacian is a family of operators, indexed by $b \in {\bf R}^{\ast}_{+}$, acting on the total space of the tangent bundle of a Riemannian manifold, that interpolates between the ordinary Laplacian as $b \rightarrow 0$ and the generator of the geodesic flow as $b\rightarrow +\infty$. Up to lower order terms, the hypoelliptic Laplacian is a geometric version of a Fokker-Planck operator. It is not self-adjoint, it is not elliptic, it is hypoelliptic. There is a probabilistic counterpart to the hypoelliptic Laplacian, which is a 1-parameter family of differential equations, known as geometric Langevin equations, that interpolates between Brownian motion and the geodesic flow.

##### Analytic torsion and dynamical zeta function on locally symmetric spaces

The relation between the spectrum of the Laplacian and the closed geodesics on a closed Riemannian manifold is one of the central themes in differential geometry. Fried conjectured that the analytic torsion, which is an alternating product of regularized determinants of the Hodge Laplacians, equals the zero value of the dynamical zeta function. In the first part of the talk, we will give a formal proof of this conjecture based on the path integral and Bismut-Goette's V-invariants. In the second part, we will give the rigorous arguments in the case where the underlying manifold is a closed locally symmetric space. The proof is based on the Bismut's formula for semisimple orbital integrals.

##### Mean convex level set flow in general ambient manifolds

Mean curvature flow is a geometric heat equation for hyper-surfaces, which is the gradient flow of the surface area functional. The flow typically becomes singular at finite time, after which it can be extended by an object called the "level set flow". In general, the level set flow is not that well behaved, but in the important mean convex case, where the initial hypersurface is a boundary of a domain which starts moving inward, a beautiful regularity and structure theory was developed in the last 20 years by Brian White. While parts of this theory work in full generality, parts were only known to hold in either the Euclidean setting or in low dimensions. We prove two new estimates for the level set flow of mean convex domains in general Riemannian manifolds.

##### New developments in strongly positive and nonnegative curvature

Strongly positive curvature is an intermediate condition between positive-definiteness of the curvature operator and positive sectional curvature, defined in terms of modifying the curvature operator with a 4-form to make it positive-definite. In this talk, I will discuss some background and two recent results on the subject: the classification of homogeneous spaces with strongly positive curvature, and the verification that all manifolds known to admit metrics with nonnegative sectional curvature also admit metrics with strongly nonnegative curvature. I will also report on work in progress using the Bochner technique in this context to find new topological obstructions. This is based on joint work with R. Mendes (WWU Muenster).

##### On the geometry of graph manifolds

In many geometric problems the curvature tensor of a Riemannian manifold has large nullity. We show that, under certain regularity assumptions, a Riemannian manifold with almost maximal nullity is isometric to an n-dimensional graph manifold. As a consequence we show that Nomizu's conjecture holds for finite volume manifolds. This is joint work with Luis Florit.

##### Infinitely many solutions to the Yamabe problem on noncompact manifolds

**Please note special time. **I will discuss the existence of infinitely many complete metrics with constant positive scalar curvature on prescribed conformal classes on certain noncompact product manifolds. These include products of closed manifolds with constant positive scalar curvature and simply-connected symmetric spaces of noncompact or Euclidean type; in particular, $S^m \times R^d$ and $S^m \times H^d$. As a consequence, one obtains infinitely many periodic solutions to the singular Yamabe problem on $S^m \setminus S^k$, for all $0 \leq k < (m - 2)/2$. I will also show that all Bieberbach groups are periods of bifurcating branches of solutions to the Yamabe problem on $S^m \times R^d$.This is a joint work with R. Bettiol, UPenn.

##### Static three-manifolds with positive scalar curvature

A Riemannian manifold is called static when it admits a non-trivial solution to a second-order equation that naturally appears both in Geometry (e.g, in the problem of prescribing the scalar curvature) and Physics (e.g., in the study of static black-holes). In this talk we will explain some results towards the classification of static three-manifolds with positive scalar curvature. In these results, we employ minimal surfaces and also explore the correspondence between static three-manifolds manifolds and Einstein four-manifolds (possibly with edge-cone singularities) admitting an isometric circle action of a certain type.

##### Boundary effect of scalar curvature

In this talk, we will discuss the effect of the scalar curvature of a Riemannian metric of a compact 3-manifold on the boundary geometry of the manifold. In particular, we will demonstrate that, on any compact Riemannian 3-manifold with nonnegative scalar curvature, if the boundary is a topological 2-sphere with nonnegative mean curvature, then the total mean curvature of the boundary is bounded from above by a constant depending only on the induced metric on the boundary. As an application, we propose a variational analogue of the Brow-York quasi-local mass functional in general relativity. This is a joint work with Christos Mantoulidis.

##### Volumes of minimal hypersurfaces and stationary geodesic nets

We will prove an upper bound for the volume of a minimal hypersurface in a closed Riemannian manifold conformally equivalent to a manifold with Ric > -(n-1). In the second part of the talk we will construct a sweepout of a closed 3-manifold with positive Ricci curvature by 1-cycles of controlled length and prove an upper bound for the length of a stationary geodesic net. These are joint works with Parker Glynn-Adey (Toronto) and Xin Zhou (MIT).

##### The Dirichlet problem for the constant mean curvature equation in Sol$_3$

**Please note different day (Thursday). ** The Dirichlet problem for CMC surfaces with infinite boundary data was first studied for minimal graphs in $\mathbb{R}^3$ by Jenkins and Serrin and then by Sprück for CMC $H$ surfaces. The existence results, which consist in giving necessary and sufficient conditions on a domain for it to admit a solution to the problem were later obtained for different ambient spaces such as $\mathbb{H}^3,$ $\mathbb{H}^2 \times \mathbb{R},$ $\mathbb{S}^2 \times \mathbb{R},$ among others. In this talk we will describe some existence results for the Dirichlet problem in Sol$_3.$ This is a joint work with Ana Menezes.

##### Uniqueness of immersed spheres in three-manifolds. Proof of a conjecture by Alexandrov

In this talk we generalize Hopf's famous classification of constant mean curvature spheres in R^3 to the general situation of classes of surfaces modeled by arbitrary elliptic PDEs in arbitrary three-manifolds, with the only hypothesis of the existence of a family of "candidate surfaces". In this way, we prove that any immersed sphere in such a class of surfaces is a candidate sphere. Among several applications, we solve two open problems of classical surface theory; we prove a 1956 conjecture by A.D. Alexandrov on the uniqueness of immersed spheres in R^3 that satisfy a general elliptic prescribed curvature equation, and show as a consequence that round spheres are the only elliptic Weingarten spheres immersed in R^3. Joint work with Jose A. Galvez.

##### Min-max theory in Gaussian space and Entropy Conjecture

**Please note additional talk and special time (4:15). **Minimal surfaces are critical points of the area functional. The min-max theory is a variational theory for constructing saddle point type, unstable minimal surfaces. In this talk, we will introduce a min-max theory in a specific space--the Gaussian probability space. Minimal surfaces in Gaussian space are also called self-shrinkers, which model the singularities of the Mean Curvature Flow. Self-shrinkers are unstable with respect to the second variation of area. Any variational construction of self-shrinkers must be of min-max type. As an application, we will address a conjecture concerning the entropy of closed surfaces. The entropy is a quantity which measures complexity of a surface.