Geometry and topology of locally convex hypersurface

A fundamental open question posed by S. T. Yau asks when does a closed
curve in 3-space bound a positively curved surface. In this talk we
present a survey of recent results related to this problem. In
particular we show that a closed submanifold of codimension 2 in
Euclidean space bounds at most finitely many topological types of
complete hypersurfaces with nonnegative curvature, which settles a
question of Guan and Spruck. Further we discuss analogous results for
arbitrary Riemannian submanifolds. On the other hand, we show that these
finiteness theorems may not hold if the codimension is too high or the
boundary is not sufficiently regular. The proofs employ, among other
methods, the Gromov-Perelman theory of Alexandrov spaces with curvature
bounded below, and a relative version of Nash's isometric embedding
theorem. These results include joint works with Stephanie Alexander,
Robert Greene, Marek Kossowski, and Jeremy Wong.