Rob Kusner, University of Massachusetts Amherst

Abstract: Soap bubbles, and more general equilibrium fluid droplets, are modeled by constant mean curvature (CMC) surfaces embedded in 3-space. The round sphere and cylinder are the simplest examples. We explore the moduli and rigidity theory for complete CMC surfaces, especially, the class of genus zero coplanar surfaces with a finite number of ends, whose moduli space is (mostly) understood. It's naturally diffeomorphic to the product of hyperbolic 3-space with the complex affine space of normalized monic polynomials in one complex variable.

If time permits, we'll also discuss some recent work with Jeremy Leach and Rafe Mazzeo constructing genus one CMC surfaces all of whose ends are asymptotically cylindrical (answering an old question of Rick Schoen). And with any luck, expect some pictures....