The Southeast Geometry Seminar (SGS) is a semiannual series of one day event sponsored by:
005 Skiles Building
Abstract: A classical uniqueness theorem of Alexandrov says that: a closed strictly convex twice differentiable surface in R3 is uniquely determined to within a parallel translation when one gives a proper function of the principle curvatures. We will talk about a new PDE proof for this thorem by using the maximal principle and weak uniqueness continuation theorem of Bers-Nirenberg. More generally, we prove a version of this theorem with the minimal regularity assumption: the spherical hessians of the supporting functions for the corresponding convex bodies as Radon measures are nonsingular. This is a joint work with P. Guan and Z. Wang.
Abstract: We report recent progresses in our effort of seeking methods to derive a priori second order estimates for fully nonlinear elliptic (and parabolic) equations on real or complex manifolds under general structure conditions. We are concerned with both equations on closed manifolds, and the Dirichlet problem on manifolds with boundary without imposing geometric restrictions to the boundary except being smooth and compact. In particular, our existence results are essentially optimal for domains in real or complex Euclidean space (or manifolds with nonnegative curvature). In this talk we'll discuss the role of concavity and subsolution.
Abstract: Perelman's celebrated stability theorem showed that if a convergent sequence of Alexandrov spaces does not drop in dimension on passing to the limit, then the objects in the tail of the sequence are homeomorphic to the limit.
In this talk, the theorem will be extended to an equivariant setting. As an application, it will be shown that two classes of Riemannian orbifolds, defined by geometric and spectral constraints, are finite up to orbifold homeomorphism.
Abstract: Consider a liquid in contact with a solid. In equilibrium the free surface will have constant mean curvature and be subject to the volume constraint. We focus on easily visualized examples where there is a symmetry-breaking bifurcation at a critical juncture with a loss of stability as volume is increased. We follow the bifurcating family.