A Differential Perspective on Gradient Flows on CAT(κ) -Spaces and Applications

We review the theory of Gradient Flows in the framework of convex and lower semicontinuous functionals on CAT(κ) -spaces and prove that they can be characterized by the same differential inclusion yt′∈-∂-E(yt) one uses in the smooth setting and more precisely that yt′ selects the element of minimal norm in - ∂-E(yt). 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 L2 and CAT(0) valued maps: we define the Laplacian of such L2 map as the element of minimal norm in - ∂-E(u) , provided it is not empty. The theory of gradient flows ensures that the set of maps admitting a Laplacian is L2-dense. Basic properties of this Laplacian are then studied.