论文标题
准细分凸锥的仿射变形
Affine deformations of quasi-divisible convex cones
论文作者
论文摘要
对于$ \ mathrm {sl}的任何子组（3，\ mathbb {r}）\ ltimes \ mathbb {r}^3 $，通过将翻译部分添加到$ \ mathrm {slrm {Slrm {sl}的子组（3，3，\ mathbb {r}）$中，这是一个有限的shoute-withe shoute-witifuly是一个有限的confutive slime-withe shiper，slrm {slrm {slrm {slrm {slrm {slrm {sl}抛物线元素的翻译部分的自然状况，该组对$ \ Mathbb {r}^3 $的仿射作用具有不连续性的凸形域，这些凸域在某种意义上是常规的，从而使全球双曲线平面空位的混乱结果概括了。然后，我们对所有这些域进行了分类，并表明每个域的商都是由凸面表面叶状的仿射歧管，具有恒定的仿射高斯曲率。该证明是基于具有凸锥的仿射空间的几何形状与凸形管域的几何形状之间的对应关系。作为一个独立的结果，我们表明，此类组的模量空间是有限体积凸射射击结构的模量空间上的矢量束，其等级等于Teichmüller空间的尺寸。
For any subgroup of $\mathrm{SL}(3,\mathbb{R})\ltimes\mathbb{R}^3$ obtained by adding a translation part to a subgroup of $\mathrm{SL}(3,\mathbb{R})$ which is the fundamental group of a finite-volume convex projective surface, we first show that under a natural condition on the translation parts of parabolic elements, the affine action of the group on $\mathbb{R}^3$ has convex domains of discontinuity that are regular in a certain sense, generalizing a result of Mess for globally hyperbolic flat spacetimes. We then classify all these domains and show that the quotient of each of them is an affine manifold foliated by convex surfaces with constant affine Gaussian curvature. The proof is based on a correspondence between the geometry of an affine space endowed with a convex cone and the geometry of a convex tube domain. As an independent result, we show that the moduli space of such groups is a vector bundle over the moduli space of finite-volume convex projective structures, with rank equaling the dimension of the Teichmüller space.
