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 $$y_t{\textquoteright}\in -\partial ^-\textsf {E} (y_t)$$one uses in the smooth setting and more precisely that $$y_t{\textquoteright}$$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{\textendash}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.

}, isbn = {1559-002X}, url = {https://doi.org/10.1007/s12220-021-00701-5}, author = {Nicola Gigli and Francesco Nobili} } @article {2021, title = {Displacement convexity of Entropy and the distance cost Optimal Transportation}, journal = {Annales de la Facult{\'e} des sciences de Toulouse : Math{\'e}matiques}, volume = {Ser. 6, 30}, year = {2021}, pages = {411{\textendash}427}, doi = {10.5802/afst.1679}, url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1679/}, author = {Fabio Cavalletti and Nicola Gigli and Flavia Santarcangelo} } @booklet {2021, title = {A first-order condition for the independence on p of weak gradients}, year = {2021}, author = {Nicola Gigli and Francesco Nobili} } @booklet {2021, title = {Monotonicity formulas for harmonic functions in RCD(0,N) spaces}, year = {2021}, abstract = {We generalize to the\ RCD(0,N)\ setting a family of monotonicity formulas by Colding and Minicozzi for positive harmonic functions in Riemannian manifolds with non-negative Ricci curvature. Rigidity and almost rigidity statements are also proven, the second appearing to be new even in the smooth setting. Motivated by the recent work in [AFM] we also introduce the notion of electrostatic potential in\ RCD\ spaces, which also satisfies our monotonicity formulas. Our arguments are mainly based on new estimates for harmonic functions in\ RCD(K,N)\ spaces and on a new functional version of the {\textquoteleft}(almost) outer volume cone implies (almost) outer metric cone{\textquoteright} theorem.

}, author = {Nicola Gigli and Ivan Yuri Violo} } @booklet {2021, title = {Parallel transport on non-collapsed $\mathsfRCD(K,N)$ spaces}, year = {2021}, abstract = {We provide a general theory for parallel transport on non-collapsed\ RCD\ spaces obtaining both existence and uniqueness results. Our theory covers the case of geodesics and, more generally, of curves obtained via the flow of sufficiently regular time dependent vector fields: the price that we pay for this generality is that we cannot study parallel transport along a single such curve, but only along almost all of these (in a sense related to the notions of Sobolev vector calculus and Regular Lagrangian Flow in the nonsmooth setting).

The class of\ ncRCD\ spaces contains finite dimensional Alexandrov spaces with curvature bounded from below, thus our construction provides a way of speaking about parallel transport in this latter setting alternative to the one proposed by Petrunin (1998). The precise relation between the two approaches is yet to be understood.

In this paper we prove that, within the framework of $\textsf{RCD}^\star(K,N)$ spaces with $N\<\infty$, the entropic cost (i.e. the minimal value of the Schr{\"o}dinger problem) admits:A threefold dynamical variational representation, in the spirit of the Benamou{\textendash}Brenier formula for the Wasserstein distance; A Hamilton{\textendash}Jacobi{\textendash}Bellman dual representation, in line with Bobkov{\textendash}Gentil{\textendash}Ledoux and Otto{\textendash}Villani results on the duality between Hamilton{\textendash}Jacobi and continuity equation for optimal transport;A Kantorovich-type duality formula, where the Hopf{\textendash}Lax semigroup is replaced by a suitable {\textquoteleft}entropic{\textquoteright} counterpart.We thus provide a complete and unifying picture of the equivalent variational representations of the Schr{\"o}dinger problem as well as a perfect parallelism with the analogous formulas for the Wasserstein distance. Riemannian manifolds with Ricci curvature bounded from below are a relevant class of $\textsf{RCD}^*(K,N)$ spaces and our results are new even in this setting.

}, issn = {1432-2064}, doi = {10.1007/s00440-019-00909-1}, url = {https://doi.org/10.1007/s00440-019-00909-1}, author = {Nicola Gigli and Luca Tamanini} } @article {GIGLI2019, title = {Differential structure associated to axiomatic Sobolev spaces}, journal = {Expositiones Mathematicae}, year = {2019}, abstract = {The aim of this note is to explain in which sense an axiomatic Sobolev space over a general metric measure space ({\`a} la Gol{\textquoteright}dshtein{\textendash}Troyanov) induces {\textendash} under suitable locality assumptions {\textendash} a first-order differential structure.

}, keywords = {Axiomatic Sobolev space, Cotangent module, Locality of differentials}, issn = {0723-0869}, doi = {https://doi.org/10.1016/j.exmath.2019.01.002}, url = {http://www.sciencedirect.com/science/article/pii/S0723086918300975}, author = {Nicola Gigli and Enrico Pasqualetto} } @article {gigli_rigoni_2019, title = {A Note About the Strong Maximum Principle on RCD Spaces}, journal = {Canadian Mathematical Bulletin}, volume = {62}, number = {2}, year = {2019}, pages = {259{\textendash}266}, publisher = {Canadian Mathematical Society}, abstract = {We give a direct proof of the strong maximum principle on finite dimensional RCD spaces based on the Laplacian comparison of the squared distance.

}, doi = {10.4153/CMB-2018-022-9}, author = {Nicola Gigli and Chiara Rigoni} } @article {1903.04302, title = {Quasi-continuous vector fields on RCD spaces}, year = {2019}, author = {Cl{\'e}ment Debin and Nicola Gigli and Enrico Pasqualetto} } @article {1807.10063, title = {Differential of metric valued Sobolev maps}, year = {2018}, author = {Nicola Gigli and Enrico Pasqualetto and Elefterios Soultanis} } @article {1803.05374, title = {On the notion of parallel transport on RCD spaces}, year = {2018}, author = {Nicola Gigli and Enrico Pasqualetto} } @article {Gigli2018, title = {Recognizing the flat torus among RCD*(0,N) spaces via the study of the first cohomology group}, journal = {Calculus of Variations and Partial Differential Equations}, volume = {57}, number = {4}, year = {2018}, month = {Jun}, pages = {104}, abstract = {We prove that if the dimension of the first cohomology group of a $\mathsf{RCD}^\star (0,N)$ space is $N$, then the space is a flat torus. This generalizes a classical result due to Bochner to the non-smooth setting and also provides a first example where the study of the cohomology groups in such synthetic framework leads to geometric consequences.

}, issn = {1432-0835}, doi = {10.1007/s00526-018-1377-z}, url = {https://doi.org/10.1007/s00526-018-1377-z}, author = {Nicola Gigli and Chiara Rigoni} } @article {gigli2018second, title = {Second order differentiation formula on RCD(K, N) spaces}, journal = {Rendiconti Lincei-Matematica e Applicazioni}, volume = {29}, number = {2}, year = {2018}, pages = {377{\textendash}386}, author = {Nicola Gigli and Luca Tamanini} } @article {1802.02463, title = {Second order differentiation formula on RCD*(K,N) spaces}, year = {2018}, author = {Nicola Gigli and Luca Tamanini} } @article {2017arXiv170707595A, title = {The injectivity radius of Lie manifolds}, journal = {ArXiv e-prints}, year = {2017}, abstract = {We prove in a direct, geometric way that for any compatible Riemannian metric on a Lie manifold the injectivity radius is positive

}, keywords = {(58J40), 53C21, Mathematics - Differential Geometry}, url = {https://arxiv.org/pdf/1707.07595.pdf}, author = {Paolo Antonini and Guido De Philippis and Nicola Gigli} } @article {1701.03932, title = {Second order differentiation formula on compact RCD*(K,N) spaces}, year = {2017}, author = {Nicola Gigli and Luca Tamanini} } @article {1607.05188, title = {Behaviour of the reference measure on RCD spaces under charts}, year = {2016}, author = {Nicola Gigli and Enrico Pasqualetto} } @article {1611.09645, title = {Equivalence of two different notions of tangent bundle on rectifiable metric measure spaces}, year = {2016}, author = {Nicola Gigli and Enrico Pasqualetto} } @article {2013, title = {The splitting theorem in non-smooth context}, year = {2013}, abstract = {We prove that an infinitesimally Hilbertian $CD(0,N)$ space containing a line splits as the product of $R$ and an infinitesimally Hilbertian $CD(0,N -1)$ space. By {\textquoteleft}infinitesimally Hilbertian{\textquoteright} we mean that the Sobolev space $W^{1,2}(X,d,m)$, which in general is a Banach space, is an Hilbert space. When coupled with a curvature-dimension bound, this condition is known to be stable with respect to measured Gromov-Hausdorff convergence.}, url = {http://preprints.sissa.it/handle/1963/35306}, author = {Nicola Gigli} }