A differential perspective on Gradient Flows on ${\sf CAT}(\kappa)$-spaces and applications

TL;DR

This paper extends the theory of gradient flows to ${\sf CAT}(\kappa)$-spaces, characterizing them via differential inclusions and applying these results to define Laplacians on ${\sf CAT}(0)$-valued maps, with implications for analysis on metric spaces.

Contribution

It generalizes gradient flow characterizations to ${\sf CAT}(\kappa)$-spaces and introduces a Laplacian for ${\sf CAT}(0)$-valued maps based on the minimal norm element in the subdifferential.

Findings

Gradient flows characterized by differential inclusions in ${\sf CAT}(\kappa)$-spaces.

Defined Laplacian for ${\sf CAT}(0)$-valued maps via minimal norm subdifferential elements.

Set of maps admitting a Laplacian is dense in $L^2$ space.

Abstract

We review the theory of Gradient Flows in the framework of convex and lower semicontinuous functionals on ${\sf CAT}(\kappa)$-spaces and prove that they can be characterized by the same differential inclusion $y_t'\in-\partial^-{\sf E}(y_t)$ one uses in the smooth setting and more precisely that $y_t'$ selects the element of minimal norm in $-\partial^-{\sf E}(y_t)$. This generalizes previous results in this direction where the energy was also assumed to be Lipschitz. We then apply such result to the Korevaar-Schoen energy functional on the space of $L^2$ and ${\sf CAT}(0)$ valued maps: we define the Laplacian of such $L^2$ map as the element of minimal norm in $-\partial^-{\sf E}(u)$, provided it is not empty. The theory of gradient flows ensures that the set of maps admitting a Laplacian is $L^2$-dense. Basic properties of this Laplacian are then studied.