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

1 aGigli, Nicola1 aNobili, Francesco uhttps://doi.org/10.1007/s12220-021-00701-500414nas a2200097 4500008004100000245007200041210006900113100001800182700002200200856009400222 2021 eng d00aA first-order condition for the independence on p of weak gradients0 afirstorder condition for the independence on p of weak gradients1 aGigli, Nicola1 aNobili, Francesco uhttps://www.math.sissa.it/publication/first-order-condition-independence-p-weak-gradients00401nas a2200109 4500008004100000245005500041210005200096100002200148700002400170700001800194856007900212 2021 eng d00aOn master test plans for the space of BV functions0 amaster test plans for the space of BV functions1 aNobili, Francesco1 aPasqualetto, Enrico1 aSchultz, Timo uhttps://www.math.sissa.it/publication/master-test-plans-space-bv-functions36575nas a2200109 45000080041000002450109000412100069001505203607300219100002236292700002236314856012936336 2021 eng d00aRigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds0 aRigidity and almost rigidity of Sobolev inequalities on compact 3 a

We prove that if M is a closed n-dimensional Riemannian manifold, n≥3, with Ric≥n−1 and for which the optimal constant in the critical Sobolev inequality equals the one of the n-dimensional sphere Sn, then M is isometric to Sn. An almost-rigidity result is also established, saying that if equality is almost achieved, then M is close in the measure Gromov-Hausdorff sense to a spherical suspension. These statements are obtained in the RCD-setting of (possibly non-smooth) metric measure spaces satisfying synthetic lower Ricci curvature bounds.1 aNobili, Francesco1 aViolo, Ivan, Yuri uhttps://www.math.sissa.it/publication/rigidity-and-almost-rigidity-sobolev-inequalities-compact-spaces-lower-ricci-curvature

An independent result of our analysis is the characterization of the best constant in the Sobolev inequality on any compact CD space, extending to the non-smooth setting a classical result by Aubin. Our arguments are based on a new concentration compactness result for mGH-converging sequences of RCD spaces and on a Polya-Szego inequality of Euclidean-type in CD spaces.

As an application of the technical tools developed we prove both an existence result for the Yamabe equation and the continuity of the generalized Yamabe constant under measure Gromov-Hausdorff convergence, in the RCD-setting.