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23rd Annual Geometry Festival

Duke University, Durham, NC
April 25-27, 2008


Michael Anderson,   Conformally compact Einstein metrics with prescribed conformal infinity
We will discuss results obtained over the past few years on the existence and uniqueness of complete conformally compact (or asympotically hyperbolic) Einstein metrics which realize a given conformal structure on the boundary at infinity. While a fair amount is now known in dimension 4, we will also present a number of open problems in the area.
Robert Bryant,   Riemannian Submersions as PDE
The problem of determining the Riemannian submersions

f: (Q,g) -> (M,h)

with a given source Riemannian manifold (Q,g) will be discussed as a PDE problem.  When the dimension of the target M is greater than 1, this is an overdetermined PDE system whose local nature is not well-understood.  I will describe what is known and give some new results that classify such submersions when (Q,g) is a Riemannian space form of low dimension.
Greg Galloway,   Stability of marginally trapped surfaces with applications to black holes
A basic step in the classical black hole uniqueness theorems is Hawking's theorem on the topology of black holes, which asserts that cross sections of the event horizon in (3+1)-dimensional asymptotically flat stationary black hole spacetimes obeying the dominant energy condition are topologically 2-spheres. Recent interest and developments in the study of higher dimensional black holes has drawn attention to the question of what are the allowable black hole topologies in higher dimensions. We have addressed this question in two recent papers, the first with Rick Schoen, resulting in a natural generalization of Hawking's theorem to higher dimensions. In this talk we discuss these works and some further related developments. The results we describe are based on the geometry of marginally outer trapped surfaces, which are natural spacetime analogues of minimal surfaces.
Marcus Khuri,   The Yamabe Problem Revisited

John Lott,   Optimal transport in Riemannian geometry and Ricci flow
In 1781, Monge asked about the optimal way to transport a dirtpile from one place to another.  The optimal transport problem has had recent applications to Riemannian geometry (collapse with Ricci curvature bounded below) and Ricci flow (monotonic quantities).   I will describe these developments.
Bill Minicozzi,   The rate of change of width under flows
I will discuss a geometric invariant, that we call the width, of a manifold and first show how it can be realized as the sum of areas of minimal 2-spheres. When M is a homotopy 3-sphere, the width is loosely speaking the area of the smallest 2-sphere needed to ``pull over'' M. Second, we will estimate the rate of change of width under various geometric flows to prove sharp estimates for extinction times. This is joint work
with Toby Colding.
Duong Phong,   Stability and constant scalar curvature

Jeff Viaclovsky,   Orthogonal Complex Structures
I will discuss various aspects of orthogonal complex structures on domains in Euclidean spaces, and the connection with twistor theory. This is joint work with Simon Salamon and Lev Borisov.

Back to the 23rd
East Coast Geometry Festival
Web page:  Duke Math Department
Comments to:  geomfest@math.duke.edu
Started: Oct 31, 2007.    Updated: Mar 19, 2008.
URL: http://www.math.duke.edu/conferences/geomfest08/abstracts.html