Deformations of unbounded convex bodies and hypersurfaces

2009/12/15 by Mohammad Ghomi, Ghomi, Mohammad
Mathematics · #52A20 #53A07 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometry and complex manifolds #Metric Geometry (math.MG) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.0912.2780

openalex publication_date 2009/12/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the topology of the space \d\Kⁿ of complete convex hypersurfaces of \Rⁿ which are homeomorphic to \R^n-1. In particular, using Minkowski sums, we construct a deformation retraction of \d\Kⁿ onto the Grassmannian space of hyperplanes. So every hypersurface in \d \Kⁿ 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\Kⁿ consisting of smooth, strictly convex, or positively curved hypersurfaces are each contractible, which settles a question of H. Rosenberg.

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