Quasi-homogeneous domains and convex affine manifolds

2003/05/01 by Kyeonghee Jo, Jo, Kyeonghee
Mathematics · #52A20 #57M50 #FOS: Mathematics #Geometric Topology (math.GT) #math.GT #msc:52A20 #msc:57M50

paper · pdf · doi:10.48550/arxiv.math/0305027

21 pages, 3 figures

arxiv created 2003/05/01 · arxiv updated 2009/11/30

Abstract

In this article, we study convex affine domains which can cover a compact affine manifold. For this purpose, we first show that every strictly convex quasi-homogeneous projective domain has at least C¹ boundary and it is an ellipsoid if its boundary is twice differentiable. And then we show that an n-dimensional paraboloid is the only strictly convex quasi-homogeneous affine domain in \mathbb Rⁿ up to affine equivalence. Furthermore we prove that if a strictly convex quasi-homogeneous projective domain is C^α on an open subset of its boundary, then it is C^α everywhere. Using this fact and the properties of asymptotic cones we find all possible shapes for developing images of compact convex affine manifolds with dimension ≤ 4.

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