
23rd Annual
Geometry Festival
Duke University, Durham, NC
April 2527, 2008
ABSTRACTS
OF TALKS

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
wellunderstood. 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 2spheres. 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 2spheres.
When M is a homotopy 3sphere, the width is loosely speaking the
area of the smallest 2sphere 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.



