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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 rankat least $2$ with a geometric group action is rigid -- isometric to aRiemannian symmetric space, a Euclidean building, or splits as a metricproduct. This paper is the first in a series motivated by Ballmann'sconjecture. Here we prove that a CAT(0) space of rank at least $n\geq 2$ isrigid 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 reliesessentially 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 aperiodic Morse $n$-flat $F$ contains a regular point -- a point with aTits-neighborhood entirely contained in $\partial_T F$. More precisely, we showthat the set of singular points in $\partial_T F$ can be covered by finitelymany round spheres of positive codimension.

CAT(0) spaces of higher rank I