Deformations of unbounded convex bodies and hypersurfaces

TL;DR

This paper explores the topology of complete convex hypersurfaces in Euclidean space, demonstrating they can be deformed onto hyperplanes while preserving curvature properties, and shows certain subclasses are contractible.

Contribution

It introduces a deformation retraction of the space of convex hypersurfaces onto hyperplanes using Minkowski sums, and proves contractibility of subclasses like smooth or strictly convex hypersurfaces.

Findings

Deformation retraction onto the Grassmannian space of hyperplanes.

Total curvature evolves monotonically during deformation.

Subspaces of smooth, strictly convex, or positively curved hypersurfaces are contractible.

Abstract

We study the topology of the space $\d\K^n$ of complete convex hypersurfaces of $\R^n$ which are homeomorphic to $\R^{n-1}$. In particular, using Minkowski sums, we construct a deformation retraction of $\d\K^n$ onto the Grassmannian space of hyperplanes. So every hypersurface in $\d \K^n$ may be flattened in a canonical way. Further, the total curvature of each hypersurface evolves continuously and monotonically under this deformation. We also show that, modulo proper rotations, the subspaces of $\d\K^n$ consisting of smooth, strictly convex, or positively curved hypersurfaces are each contractible, which settles a question of H. Rosenberg.