具平坦欧氏边界的局部凸浸入超曲面

Locally convex immersed surfaces with flat Euclidean boundary

摘要:

在仿射微分几何中，局部强（一致）凸的浸入超曲面的几何与拓扑性质非常复杂，这导致其欧氏边界的性质同样复杂。本文构造出一类新的局部强凸的浸入超曲面，其欧氏边界是平坦的（即边界落在一个超平面内），但曲面本身却不是整体凸的，这与目前现存的结论完全不同。

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.