论文标题
CAT（0）较高等级的空间
CAT(0) spaces of higher rank I
论文作者
论文摘要
猫（0）空间的排名至少为$ n $，如果每个测量中的每个地理位置都在$ n $ flat中。鲍尔曼（Ballmann）的较高排名刚度猜想预测，与几何组动作的猫（0）排名至少$ 2 $是刚性的 - 静脉内的对称空间，欧几里得建筑物或分配为度量产品。本文是Ballmann的猜想激励的系列中的第一篇。在这里，我们证明，如果包含定期的$ n $ -flat，其cat（0）排名至少$ n \ geq 2 $是刚性的，并且其山雀边界具有尺寸$（n-1）$。这不需要几何组动作。结果基本上依赖于不绑定平坦半空间的公寓的研究 - 所谓的摩尔斯公寓。我们表明，定期摩尔斯$ n $ -flat $ f $的山雀边界$ \ partial_t f $包含一个常规点 - 一个完全包含在$ \ partial_t f $中的山雀 - 邻里的点。更确切地说，我们表明，$ \ partial_t f $中的单数点集可以被有限的正面圆形圆形覆盖。
A CAT(0) space has rank at least $n$ if every geodesic lies in an $n$-flat. Ballmann's Higher Rank Rigidity Conjecture predicts that a CAT(0) space of rank at least $2$ with a geometric group action is rigid -- isometric to a Riemannian symmetric space, a Euclidean building, or splits as a metric product. This paper is the first in a series motivated by Ballmann's conjecture. Here we prove that a CAT(0) space of rank at least $n\geq 2$ is rigid if it contains a periodic $n$-flat and its Tits boundary has dimension $(n-1)$. This does not require a geometric group action. The result relies essentially on the study of flats which do not bound flat half-spaces -- so-called Morse flats. We show that the Tits boundary $\partial_T F$ of a periodic Morse $n$-flat $F$ contains a regular point -- a point with a Tits-neighborhood entirely contained in $\partial_T F$. More precisely, we show that the set of singular points in $\partial_T F$ can be covered by finitely many round spheres of positive codimension.
