A Differential Perspective on Gradient Flows on $$\textsf {CAT} (\kappa )$$-Spaces and Applications

Abstract

We review the theory of Gradient Flows in the framework of convex and lower semicontinuous functionals on $$\textsf {CAT} (\kappa )$$ CAT ( κ ) -spaces and prove that they can be characterized by the same differential inclusion $$y_t'\in -\partial ^-\textsf {E} (y_t)$$ y t ′ ∈ - ∂ - E ( y t ) one uses in the smooth setting and more precisely that $$y_t'$$ y t ′ selects the element of minimal norm in $$-\partial ^-\textsf {E} (y_t)$$ - ∂ - 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 $$L^2$$ L 2 and (0) valued maps: we define the Laplacian of such $$L^2$$ L 2 map as the element of minimal norm in $$-\partial ^-\textsf {E} (u)$$ - ∂ - E ( u ) , provided it is not empty. The theory of gradient flows ensures that the set of maps admitting a Laplacian is $$L^2$$ L 2 -dense. Basic properties of this Laplacian are then studied.

Lipschitz continuity and Bochner-Eells-Sampson inequality for harmonic maps from $\mathrm{RCD}(K,N)$ spaces to $\mathrm{CAT}(0)$ spaces

“…HS m is obtained under the same assumptions. We report below the detailed computation showing that the stronger bound (2.7) actually holds and fixing a typo in [40].…”

“…HS , see for instance [60,Chapter 8]. This is generalized to the present setting in [40,Theorem 4.1]. See also [74] for the case of maps with Euclidean source and CAT(0) target.…”

A canonical infinitesimally Hilbertian structure on locally Minkowski spaces

“…while, on the other hand, Z 1 ∩ E ε (z) has a density point in z. Then, according to (12), we can deduce that…”

“…This functional was first introduced in [20], as the key tool to study the harmonicity of maps defined from a smooth manifold to a general metric space (see also [19]). We point out that recently this energy has been a central object of investigation in a series of works [13,12,14], where it was extended to maps defined on more general RCD spaces. Notice that since this new distance d ′ is defined intrinsically, it will be in particular canonical, meaning that it will depend only on the space and not on particular choices of some other geometric objects.…”

On the regularity of harmonic maps from ${\sf RCD}(K,N)$ to ${\sf CAT}(0)$ spaces and related results

“…Notice in particular that the key 'subpartition lemma' from [60] may fail in this framework. This motivated us to develop a research program [46,45,24,40,37], in collaboration with a diverse set of coauthors, aimed at generalizing the above to its natural framework of RCD spaces. In particular, in [46], suitably adapting Korevaar-Schoen's approach, we showed that the Dirichlet problem is well posed in this framework, so that at least the concept of 'harmonic map from a domain in an RCD(K, N ) space to a CAT(0) one' is well defined.…”