Infinitesimal Hilbertianity of Locally $$\mathrm{CAT}(\kappa )$$-Spaces

Di Marino, Simone; Gigli, Nicola; Pasqualetto, Enrico; Soultanis, Elefterios

doi:10.1007/s12220-020-00543-7

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Journal article

Infinitesimal Hilbertianity of Locally $$\mathrm{CAT}(\kappa )$$-Spaces

Di Marino, Simone Istituto Nazionale di Alta Matematica, Unità INdAM SNS Pisa, Piazza dei Cavalieri 7, 56126 Pisa, Italy
Gigli, Nicola SISSA, Via Bonomea 265, 34136 Trieste, Italy
Pasqualetto, Enrico SISSA, Via Bonomea 265, 34136 Trieste, Italy - Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), 40014 Jyväskylä, Finland
Soultanis, Elefterios SISSA, Via Bonomea 265, 34136 Trieste, Italy - University of Fribourg, Chemin du Musee 23, 1700 Fribourg, Switzerland

06.11.2020

Published in:

The Journal of Geometric Analysis. - 2021, vol. 31, no. 8, p. 7621-7685

English We show that, given a metric space $$(\mathrm{Y},\textsf {d} )$$of curvature bounded from above in the sense of Alexandrov, and a positive Radon measure $$\mu $$on $$\mathrm{Y}$$giving finite mass to bounded sets, the resulting metric measure space $$(\mathrm{Y},\textsf {d} ,\mu )$$is infinitesimally Hilbertian, i.e. the Sobolev space $$W^{1,2}(\mathrm{Y},\textsf {d} ,\mu )$$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 $$x\in \mathrm{Y}$$is the tangent cone at x of $$\mathrm{Y}$$. The conclusion then follows from the fact that for every $$x\in \mathrm{Y}$$such a cone is a $$\mathrm{CAT}(0)$$space and, as such, has a Hilbert-like structure.

Faculty

Faculté des sciences et de médecine

Department

Département de Mathématiques

Language