Convexity of nonnegatively curved hypersurfaces with free boundary on a sphere

In 1897 Hadamard proved that any closed immersed surface with positive Gaussian curvature in 3-dimensional Euclidean space must be the boundary of a convex body. After the later efforts by Stoker, van Heijenoort, Chern-Lashof and Sacksteder, the problem when hypersurfaces without boundary are globally convex now is quite clear. In this talk we shall introduce our recent work on the convexity problem for hypersurfaces with boundary. More precisely, we prove that any compact immersed hypersurface in Euclidean space with nonnegative sectional curvatures and with free boundary on the standard sphere must be globally convex. The key ingredient in the proof is a gluing process which reduces the problem with boundary to that without boundary. This work is joint with Mohammad Ghomi.