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 Schrödinger problem) admits:A threefold dynamical variational representation, in the spirit of the Benamou–Brenier formula for the Wasserstein distance; A Hamilton–Jacobi–Bellman dual representation, in line with Bobkov–Gentil–Ledoux and Otto–Villani results on the duality between Hamilton–Jacobi and continuity equation for optimal transport;A Kantorovich-type duality formula, where the Hopf–Lax semigroup is replaced by a suitable `entropic' counterpart.We thus provide a complete and unifying picture of the equivalent variational representations of the Schrödinger 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.

UR - https://doi.org/10.1007/s00440-019-00909-1 ER - TY - JOUR T1 - Differential structure associated to axiomatic Sobolev spaces JF - Expositiones Mathematicae Y1 - 2019 A1 - Nicola Gigli A1 - Enrico Pasqualetto KW - Axiomatic Sobolev space KW - Cotangent module KW - Locality of differentials AB -The aim of this note is to explain in which sense an axiomatic Sobolev space over a general metric measure space (à la Gol’dshtein–Troyanov) induces – under suitable locality assumptions – a first-order differential structure.

UR - http://www.sciencedirect.com/science/article/pii/S0723086918300975 ER - TY - JOUR T1 - A Note About the Strong Maximum Principle on RCD Spaces JF - Canadian Mathematical Bulletin Y1 - 2019 A1 - Nicola Gigli A1 - Chiara Rigoni AB -We give a direct proof of the strong maximum principle on finite dimensional RCD spaces based on the Laplacian comparison of the squared distance.

PB - Canadian Mathematical Society VL - 62 ER - TY - RPRT T1 - Quasi-continuous vector fields on RCD spaces Y1 - 2019 A1 - Clément Debin A1 - Nicola Gigli A1 - Enrico Pasqualetto ER - TY - RPRT T1 - Differential of metric valued Sobolev maps Y1 - 2018 A1 - Nicola Gigli A1 - Enrico Pasqualetto A1 - Elefterios Soultanis ER - TY - RPRT T1 - On the notion of parallel transport on RCD spaces Y1 - 2018 A1 - Nicola Gigli A1 - Enrico Pasqualetto ER - TY - JOUR T1 - Recognizing the flat torus among RCD*(0,N) spaces via the study of the first cohomology group JF - Calculus of Variations and Partial Differential Equations Y1 - 2018 A1 - Nicola Gigli A1 - Chiara Rigoni AB -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.

VL - 57 UR - https://doi.org/10.1007/s00526-018-1377-z ER - TY - JOUR T1 - Second order differentiation formula on RCD(K, N) spaces JF - Rendiconti Lincei-Matematica e Applicazioni Y1 - 2018 A1 - Nicola Gigli A1 - Luca Tamanini VL - 29 ER - TY - RPRT T1 - Second order differentiation formula on RCD*(K,N) spaces Y1 - 2018 A1 - Nicola Gigli A1 - Luca Tamanini ER - TY - JOUR T1 - The injectivity radius of Lie manifolds JF - ArXiv e-prints Y1 - 2017 A1 - Paolo Antonini A1 - Guido De Philippis A1 - Nicola Gigli KW - (58J40) KW - 53C21 KW - Mathematics - Differential Geometry AB -We prove in a direct, geometric way that for any compatible Riemannian metric on a Lie manifold the injectivity radius is positive

UR - https://arxiv.org/pdf/1707.07595.pdf ER - TY - RPRT T1 - Second order differentiation formula on compact RCD*(K,N) spaces Y1 - 2017 A1 - Nicola Gigli A1 - Luca Tamanini ER - TY - RPRT T1 - Behaviour of the reference measure on RCD spaces under charts Y1 - 2016 A1 - Nicola Gigli A1 - Enrico Pasqualetto ER - TY - RPRT T1 - Equivalence of two different notions of tangent bundle on rectifiable metric measure spaces Y1 - 2016 A1 - Nicola Gigli A1 - Enrico Pasqualetto ER - TY - RPRT T1 - The splitting theorem in non-smooth context Y1 - 2013 A1 - Nicola Gigli AB - 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 ‘infinitesimally Hilbertian’ 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. UR - http://preprints.sissa.it/handle/1963/35306 U1 - 35613 U2 - Mathematics U4 - 1 U5 - MAT/05 ER -