论文标题
关于全球性低下的阿贝尔动作及其在同质空间上的存在
On globally hypoelliptic abelian actions and their existence on homogeneous spaces
论文作者
论文摘要
我们将全球性低纤维化平滑$ \ Mathbb r^k $动作定义为动作，其沿轨道叶叶叶叶叶片是全球性低纤维化差异操作员。当$ k = 1 $时，Greenfield-Wallach和Katok对此类行动的强烈全球刚度是：每一个这样的动作都可以平稳地与圆环上的双苯胺流相结合。对同质空间上的所有均匀流\ cite {ffrh}上的所有均匀流都得到了证实。在本文中，我们推测，在均质空间中，全球范围内的均质空间中仅存在于Nilmanifolds上的同质$ \ Mathbb r^k $ Actions（$ k \ ge 2 $）。我们对这种猜想获得了部分结果：我们表明全球性低纤维化$ \ Mathbb r^2 $在同质空间上的动作$ g/γ$，其中至少有一个准单位生成器，其中$ g = sl = sl（n，\ sathbb r）$。我们还表明，对Solvmanifolds的相同类型的动作与对Nilmanifolds的均匀作用平稳结合。
We define globally hypoelliptic smooth $\mathbb R^k$ actions as actions whose leafwise Laplacian along the orbit foliation is a globally hypoelliptic differential operator. When $k=1$, strong global rigidity is conjectured for such actions by Greenfield-Wallach and Katok: every such action is smoothly conjugate to a Diophantine flow on the torus. The conjecture has been confirmed for all homogeneous flows on homogeneous spaces \cite{FFRH}. In this paper we conjecture that among homogeneous $\mathbb R^k$ actions ($k\ge 2$) on homogeneous spaces globally hypoelliptic actions exist only on nilmanifolds. We obtain a partial result towards this conjecture: we show non-existence of globally hypoelliptic $\mathbb R^2$ actions on homogeneous spaces $G/Γ$, with at least one quasi-unipotent generator, where $G= SL(n, \mathbb R)$. We also show that the same type of actions on solvmanifolds are smoothly conjugate to homogeneous actions on nilmanifolds.
