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Register a new userInfinitesimal Hilbertianity of locally CAT($kappa$)-spaces
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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.
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We prove that sub-Riemannian manifolds are infinitesimally Hilbertian (i.e., the associated Sobolev space is Hilbert) when equipped with an arbitrary Radon measure. The result follows from an embedding of metric derivations into the space of square-integrable sections of the horizontal bundle, which we obtain on all weighted sub-Finsler manifolds. As an intermediate tool, of independent interest, we show that any sub-Finsler distance can be monotonically approximated from below by Finsler ones. All the results are obtained in the general setting of possibly rank-varying structures.
We prove metric differentiation for differentiability spaces in the sense of Cheeger. As corollaries we give a new proof that the minimal generalized upper gradient coincides with the pointwise Lipschitz constant for Lipschitz functions on PI spaces, a proof that the Lip-lip constant of any Lip-lip space in the sense of Keith is equal to $1$, and new nonembeddability results.
We analyze weak convergence on $CAT(0)$ spaces and the existence and properties of corresponding weak topologies.
We construct examples of smooth 4-dimensional manifolds M supporting a locally CAT(0)-metric, whose universal cover X satisfy Hruskas isolated flats condition, and contain 2-dimensional flats F with the property that the boundary at infinity of F defines a nontrivial knot in the boundary at infinity of X. As a consequence, we obtain that the fundamental group of M cannot be isomorphic to the fundamental group of any Riemannian manifold of nonpositive sectional curvature. In particular, M is a locally CAT(0)-manifold which does not support any Riemannian metric of nonpositive sectional curvature.
We give another proof of Toyodas theorem that describes 5-point subpaces in CAT(0) length spaces
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