Infinitesimal Hilbertianity of Locally CAT(κ)-Spaces
Di Marino, S., Gigli, N., Pasqualetto, E., & Soultanis, E. (2021). Infinitesimal Hilbertianity of Locally CAT(κ)-Spaces. Journal of Geometric Analysis, 31(8), 7621-7685. https://doi.org/10.1007/s12220-020-00543-7
Published inJournal of Geometric Analysis
© The Author(s) 2020
We show that, given a metric space (Y,d)(Y,d) of curvature bounded from above in the sense of Alexandrov, and a positive Radon measure μμ on YY giving finite mass to bounded sets, the resulting metric measure space (Y,d,μ)(Y,d,μ) is infinitesimally Hilbertian, i.e. the Sobolev space W1,2(Y,d,μ)W1,2(Y,d,μ) 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∈Yx∈Y is the tangent cone at x of YY. The conclusion then follows from the fact that for every x∈Yx∈Y such a cone is a CAT(0)CAT(0) space and, as such, has a Hilbert-like structure.
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Additional information about fundingThis research has been supported by the MIUR SIR-Grant ‘Nonsmooth Differential Geometry’ (RBSI147UG4). Open access funding provided by Scuola Internazionale Superiore di Studi Avanzati – SISSA within the CRUI-CARE Agreement.
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