The Space of Spaces: Curvature Bounds and Gradient Flows on the Space of Metric Measure Spaces

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Abstract

Equipped with the L 2 , q L^{2,q} -distortion distance \DD _{2,q}, the space \XX _{2q} of all metric measure spaces (X,\d ,\m ) is proven to have nonnegative curvature in the sense of Alexandrov. Geodesics and tangent spaces are characterized in detail. Moreover, classes of semiconvex functionals and their gradient flows on \ol \XX _{2q} are presented.