Dmitry Burago, Penn State University
Gaussian Measures
of Surfaces, Ellipticity of Volume
Functionals, and Volume/Distance Estimates
Wednesday June 5, 2002, 16:00, Schreiber Bldg, Room 210.
We will discuss the following topics, which happen
to be closely related:
1. What are the restrictions on the Gaussian measures of closed surfaces,
or of surfaces with planar boundaries (the Gaussian measure of a surface
is the push-forward of the surface area under the Gaussian map to the Grassmannian).
For instance, under what conditions can one construct a
polyhedron (for instance, a two-dimensional polyhedral surface in $R^4$)
with given areas and directions of faces and no boundary (or a boundary
lying in a given two-dimensional plane)?
2. What is the relationship between different types of ellipticily for
surface area functionals (an area functional is elliptic over $Z$
(resp. $R$) if regions in affine planes are area minimizers among
all Lipschitz chains over $Z$ ($R$) with the same boundary).
3. Optimal fillings: metrics on a manifold (with boundary) that admit no
volume decreasing perturbations that do not decrease distances between boundary
points. In other words, we will be looking for situations when an inequality
for boundary distance functions of two metrics implies corresponding inequality
for the volumes.
4. Asymptotic growth of volume for large balls in a periodic metric (a
metric that admits a co-compact isometric action of an abelian group).