Student Geometry and Analysis Seminar 2009-2010

We will talk about the bubbling convergence for harmonic maps given by Sacks and Uhlenbeck in 1980's. We will begin by elementary concepts for harmonic maps. Then we will give some analytical properties which are essential in the bubbling convergence. Finally, we will construct the bubbling convergence procedure. Time permitted, we will talk about energy conservation in this convergence.

In the context of optimal transportation theory, the 2-Wasserstein distance measure the optimal transport cost between two measures on a given Riemannian manifold. In 2008, Robert McCann and Peter Topping proved that if we evolve the measures by forward heat equations and the Riemannian metric by the backward Ricci flow, the 2-Wasserstein distance is monotonically decreasing. Later in 2009, based on the same idea of the proof of this monotonicity result, Brendle, Lott, McCann, Topping, et al were able to recover several Perelman’s influential theorem via optimal transportation theory, the most notable one is the local non-collapsing theorem which is crucial to the complete proof of the Geometrization conjecture. In this talk, I will give the proof of McCann-Topping’s monotonicity result on the 2-Wasserstein distance, and discuss the interplay between Perelman’s local non-collapsing theorem and optimal transportation theory.

"Positively curved spaces" have been the central object of study since the birth of Riemannian Geometry. In the 80's, a new notion of curvature, called isotropic curvature, was introduced by Micallef and Moore. This is used in the recent proof of the differentiable sphere theorem by Schoen and Brendle. I will define this new curvature quantity and discuss its relation with minimal surfaces. I will also talk about some results and conjectures on the topology of spaces with positive isotropic curvature (PIC).

Let $(M, g)$ be a closed Riemann surface, $(N, h)$ a Riemannian manifold, and let $\phi : M \to N$ be a homeomorphism. There is a functional on the homotopy class of $\phi$, the energy functional, that takes a map $f$ and computes the integral of the square norm of its differential. Critical points of this functional are called harmonic maps. In the case that $ M $ and $ N $ are 2-dimensional, compact surfaces, minimizers of this functional are smooth, and given a homeomorphism $ f : M \to N $, there is a harmonic diffeomorphism that is homotopic to $ f $. If $ N $ is negatively curved, or if the genus of $ N $ is greater than $ 1 $, this harmonic map is the unique in the homotopy class of $ f $. In this talk, I'll show you how to prove the old theorems above in the case of negatively curved targets, and then I'll discuss how to extend those techniques to prove the exact same statements for target manifold with conic singularities. The argument is a perturbative one, so linear PDE is the main tool. In the conic setting, the relevant PDE lies in a class of linear operators called $b$-operators, whose study, originally undertaken by Melrose and Mendoza in 1983, provides us with a complete set of tools for solving this problem. We will discuss these tools and how they apply in this situation.

Many minimal submanifolds cannot be locally represented as the graphs of single-valued functions, but rather as graphs of multivalued functions. Thus it is important to study multivalued functions solving elliptic partial differential equations. An important simple case to consider is the case of 2-valued symmetric functions, that is functions of the form {-psi(x), psi(x)} at each point x, solving the Laplace equation. I will show how these functions can be studied using Almgren's frequency function.

Smoothly conformally compact metrics (or 0-metrics) are interesting from the point of view of analysis, geometry and physics. These metrics are complete metrics on the interior of a compact manifold with boundary whose sectional curvature tends to -1 upon approach to the boundary. As such they provide a model for asymptotically hyperbolic geometry. In this talk I will review some of the basics of asymptotically hyperbolic geometry and discuss recent work characterizing these metrics intrinsically. I plan to survey some recent results in this subject by Hu, Qing and Shi.

Infinite-dimensional manifolds are a nightmare for those who are addicted to writing everything in coordinates. But if you are willing to take the plunge into invariant notation, many familiar theorems from differential geometry carry over to separable Hilbert manifolds. There are, however, some notable theorems that do not work in infinite dimensions, such as the Hopf-Rinow theorem. Depending on your perspective, this is either an enormous headache or an employment opportunity. Complicating matters further, there are two kinds of Riemannian metrics in infinite dimensions - called "strong" and "weak" - just as there are two kinds of norms, complete and incomplete, on infinite-dimensional vector spaces. We'll survey these facts and, time permitting, also some facts on the even more difficult realm of Frechet manifolds, because as we'll see, sometimes even Hilbert manifolds just don't do the trick.