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≥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 ∂_TF of a periodic Morse n-flat F contains a regular point – a point with a Tits-neighborhood entirely contained in ∂_TF. More precisely, we show that the set of singular points in ∂_TF can be covered by finitely many round spheres of positive codimension.

如果每个测地线都位于n平面上，则CAT(0) 空间的秩至少为n。鲍尔曼的高阶刚性猜想预测，具有几何群作用的秩至少为 2 的 CAT(0) 空间是刚性的– 与黎曼对称空间、欧几里得建筑或作为度量积分裂等距。本文是由鲍尔曼猜想引发的系列论文中的第一篇。在这里，我们证明，如果秩至少为n ≥2 的 CAT(0) 空间包含周期性n平坦且其 Tits 边界具有维度 ( n −1)，则该空间是刚性的。这不需要几何群作用。该结果主要依赖于对不限制平坦半空间的平坦区域（即所谓的莫尔斯平坦区域）的研究。我们证明了周期性 Morse n -flat F的Tits 边界∂ _T F包含一个规则点——一个 Tits 邻域完全包含在∂ _T F中的点。更准确地说，我们证明∂ _T F中的奇点集可以被有限多个正余维的圆球覆盖。