Quasi-homogeneous domains and convex affine manifolds

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 C 1 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 R n 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 ≤ 4.2000 Mathematics Subject Classification. 52A20, 57M50.

Domains With Flat Boundary Pieces

“…As we can see in the proof of Theorem 1.2, {ξ} is either the kernel Ker(g) or the g i -image of Ker(g) for some i. Since Ran(g) ∩ R n contains F , Ker(g) is a subset of RP n−1 ∞ by Lemma 3.5 of [2]. This implies that ξ ∈ RP n−1 ∞ because g i preserves R n for all i.…”

Asymptotic Foliations of Quasi-Homogeneous Convex Affine Domains

“…The following theorem, which was proved in [2], is very important to understand quasi-homogeneous domains.…”

“…The author has proved in [2] that S(Ω) is the set of all the extreme points, and furthermore S(Ω) is the whole boundary of Ω and Aut(Ω) acts transitively on the boundary if Ω is a strictly convex domain. In this paper, we extend this result to an arbitrary properly convex domain: Theorem 1.…”

A Construction of Strictly Increasing Continuous Singular Functions

“…But there are even strictly increasing continuous singular functions (see, for example, [4] and [5]). It's well known that all the derivatives of the boundary functions of strictly convex divisible (or quasihomogeneous) projective domains are such functions if the domain is not an ellipse (see [1]). …”