UMN Geometric Analysis seminar

Abstracts:
Bob Gulliver (University of Minnesota), ' Regularity of Soap Film-Like surfaces and Total Curvature of Graphs'. A graph in a manifold M^n degree of the vertex. In a physical experiment, it is possible to dip a wire framework into soap solution and withdraw it: a surface may be formed in which three pieces of soap film meet with equal angles along edges, and where six pieces of soap film may meet symmetrically at a point. Jean Taylor in 1976 used a certain class of rectifiable sets in R^3, which she called soap film-like surfaces, as a model for such physical surfaces, and proved their interior regularity. The existence had been established earlier by Fred Almgren. We extend recent work of Choe and the speaker on branched minimal surfaces to this case (with an extra assumption of boundary regularity), proving that the density of a soap film-like surface may be controlled by the total curvature of its boundary graph. The contribution to the total curvature at a vertex is defined as the supremum over p of the sum of the complements of the interior angles to a cone over the graph with vertex p in R^n. This is joint work with Sumio Yamada.

David Dumas (Harvard), 'Grafting and Complex Projective Riemann Surfaces'. Grafting is a cut-and-paste procedure for modifying Riemann surfaces by inserting Euclidean strips along hyperbolic geodesics. A construction of Thurston shows that grafting is closely related to the theory of CP^1 structures, which were traditionally studied using complex-analytic techniques. We discuss new results comparing these two perspectives, including a geometric compactification of the space of CP^1-structures that is compatible with the foliation by leaves of constant conformal structure.

Scot Adams (UMN). 'From Lozentzian Dynamics to the Decay of Matrix Coefficients'. The Howe-Moore theorem states that any ergodic action of a connected noncompact, finite-center, simple Lie group is mixing. Thus, for example, ergodicity inherits to all noncompact closed subgroups, a very useful fact which immediately yields ergodicity of many geometrically-motivated actions. The proofs I know of Howe-Moore go via unitary representation theory. By thinking of a Hilbert space as an infinite-dimensional Riemannian manifold and by adjusting techniques originally used in studying Lorentzian (and Riemannian) dynamics, we can obtain a version of Howe-Moore that is valid for all connected Lie groups, though only useful for nonAbelian groups. Specifically, for any connected Lie group G, for any faithful irreducible unitary representation of G, we have: any matrix coefficient tends to zero, as Ad(g) leaves compact subsets of GL(\frak g), where \frak g is the Lie algebra of G.