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

Abstract

We review the theory of Gradient Flows in the framework of convex and lower semicontinuous functionals on $\mathsf{CAT}(\kappa )$-spaces and prove that they can be characterized by the same differential inclusion $y_t^{\prime }\in -{\partial }^{-}\mathsf{E}(y_t)$ one uses in the smooth setting and more precisely that $y_t^{\prime }$ selects the element of minimal norm in $-{\partial }^{-}\mathsf{E}(y_t)$. 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 $\mathsf{CAT}(0)$ valued maps: we define the Laplacian of such $L^2$ map as the element of minimal norm in $-{\partial }^{-}\mathsf{E}(u)$, 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.