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Register a new userWeak topology on CAT(0) spaces
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We analyze weak convergence on $CAT(0)$ spaces and the existence and properties of corresponding weak topologies.
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We give another proof of Toyodas theorem that describes 5-point subpaces in CAT(0) length spaces
For a k-flat F inside a locally compact CAT(0)-space X, we identify various conditions that ensure that F bounds a (k+1)-dimensional half flat in X. Our conditions are formulated in terms of the ultralimit of X. As applications, we obtain (1) constraints on the behavior of quasi-isometries between tocally compact CAT(0)-spaces, (2) constraints on the possible non-positively curved Riemannian metrics supported by certain manifolds, and (3) a correspondence between metric splittings of a complete, simply connected, non-positively curved Riemannian manifold and the metric splittings of its asymptotic cones. Furthermore, combining our results with the Ballmann, Burns-Spatzier rigidity theorem and the classical Mostow rigidity theorem, we also obtain (4) a new proof of Gromovs rigidity theorem for higher rank locally symmetric spaces.
We show that, given a metric space $(Y,d)$ of curvature bounded from above in the sense of Alexandrov, and a positive Radon measure $mu$ on $Y$ giving finite mass to bounded sets, the resulting metric measure space $(Y,d,mu)$ is infinitesimally Hilbertian, i.e. the Sobolev space $W^{1,2}(Y,d,mu)$ is a Hilbert space. The result is obtained by constructing an isometric embedding of the `abstract and analytical space of derivations into the `concrete and geometrical bundle whose fibre at $xin Y$ is the tangent cone at $x$ of $Y$. The conclusion then follows from the fact that for every $xin Y$ such a cone is a CAT(0)-space and, as such, has a Hilbert-like structure.
We introduce noncommutative weak Orlicz spaces associated with a weight and study their properties. We also define noncommutative weak Orlicz-Hardy spaces and characterize their dual spaces.
We show that the class of CAT(0) spaces is closed under suitable conformal changes. In particular, any CAT(0) space admits a large variety of non-trivial deformations.
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