On the Markus conjecture in convex case

TL;DR

This paper proves the Markus conjecture for convex affine manifolds of dimension five or less by analyzing the boundary limit sets of convex affine domains and their automorphism groups.

Contribution

It provides a positive resolution of the Markus conjecture in the convex case for low-dimensional manifolds and characterizes the boundary behavior of convex affine domains.

Findings

Convex affine domains with nonempty boundary limit sets cannot cover compact affine manifolds with parallel volume.

The Markus conjecture holds for convex affine manifolds of dimension ≤ 5.

Boundary limit set analysis is key to understanding the conjecture in the convex case.

Abstract

In this paper, we show that any convex affine domain with a nonempty limit sets on the boundary under the action of the identity component of the automorphism group cannot cover a compact affine manifold with a parallel volume, which is a positive answer to the Markus conjecture for convex case. Consequently, we show that the Markus conjecture is true for convex affine manifolds of dimension $\leq 5$.