TOPOLOGY OF NONNEGATIVELY CURVED HYPERSURFACES WITH PRESCRIBED BOUNDARY IN Rn

We prove that a smooth compact immersed submanifold of codimension 2 in R, n ≥ 3, bounds at most finitely many topologically distinct compact nonnegatively curved hypersurfaces. Analogous results for noncompact fillings are obtained as well. On the other hand, we show that these topological finiteness theorems may not hold if the prescribed boundary is not sufficiently regular, e.g., C. In particular we construct a simple closed differentiable and rectifiable curve in R which bounds infinitely many topologically distinct smooth positively curved surfaces. The proofs employ, among other methods, theorems of Gromov and Perelman on Alexandrov spaces with curvature bounded below.