Calculus of Variations and Geometric Measure Theory

S. Di Marino - N. Gigli - E. Pasqualetto - E. Soultanis

Infinitesimal Hilbertianity of locally ${\rm Cat}(\kappa)$-spaces

created by dimarino on 05 Dec 2018
modified on 06 Dec 2018


Submitted Paper

Inserted: 5 dec 2018
Last Updated: 6 dec 2018

Year: 2018

ArXiv: 1812.02086 PDF


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