Locally convex immersed surfaces with flat Euclidean boundary

Abstract:

In affine differential geometry, the geometric and topological behavior of locally strongly (uniformly) convex immersed surfaces（hypersufaces） are very complicated, so are their Euclidean boundaries. In this paper we construct a new locally strongly convex ( but not globally convex) immersed surface(hypersurfaces) with flat Euclidean boundary in $\mathbb{R}^{n+1},(n=2，3)$ , respectively, which are different from an existing conclusion.