DIFFERENTIAL GEOMETRY/PDE SEMINAR

Thursday, November 6, 2003 Padelford C-36 1:30 pm

“Building a roof out of a given set of k vectors” and ellipticity of surface area functionals

Dmitri Burago (Penn State University)

We will discuss the following topics, which happen to be closely related:

1. What are the restrictions on the Gaussian image of a closed surfaces (or of a surface with planar boundaries)? To be more precise, we are interested in weighted Gaussian image, that is in the image of the surface area under the Gauss map. For instance, under what conditions can one construct 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)? That is, given a set of k-vectors, can we build a roof?

2. Do flats minimizer surface area among all surfaces with the same boundary? Of course, they do for the surface area functional induced by Euclidean structure, but what about a normed space? A related question is: 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 Lipschits 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 implies corresponding inequality for the volumes of the metrics.

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). In other words: what is approximately the area of a ball of a large radius R in the universal cover of a Riemannian (or even Finsler) torus?

For more information about this seminar, visit the DG/PDE Seminar Web page (from the Math Department home page, www.math.washington.edu, follow the link Seminars, Colloquia, and Conferences).

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