In this paper, we show that any convex affine domain with a nonempty limit set 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 ≤ 5.

中文翻译：

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关于凸情况下的马库斯猜想

在本文中，我们证明了在自同构群的恒等分量的作用下，任何在边界上设置非空极限的凸仿射域都不能覆盖具有平行体积的紧仿射流形，这是对马库斯猜想的肯定回答对于凸情况。因此，我们证明了马库斯猜想对于维度 ≤ 5 的凸仿射流形是正确的。