When is an immersed hypersurface in Euclidean space globally convex? One answer obtained by Hadamard in 1897 is 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 answer for hypersurfaces without boundary now is quite complete. In this talk we shall focus on this problem for hypersurfaces with boundary. More precisely, our work shows 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.

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