TY - RPRT T1 - A general splitting principle on RCD spaces and applications to spaces with positive spectrum Y1 - 2023 A1 - Nicola Gigli A1 - Fabio Marconi AB -

In this paper we develop a general `analytic' splitting principle for RCD spaces: we show that if there is a function with suitable Laplacian and Hessian, then the space is (isomorphic to) a warped product. Our result covers most of the splitting-like results currently available in the literature about RCD spaces. We then apply it to extend to the non-smooth category some structural property of Riemannian manifolds obtained by Li and Wang.

ER - TY - JOUR T1 - A Differential Perspective on Gradient Flows on CAT(K)-Spaces and Applications Y1 - 2021 A1 - Nicola Gigli A1 - Francesco Nobili AB -

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.

VL - 31 SN - 1559-002X UR - https://doi.org/10.1007/s12220-021-00701-5 IS - 12 JO - The Journal of Geometric Analysis ER - TY - JOUR T1 - Displacement convexity of Entropy and the distance cost Optimal Transportation JF - Annales de la Faculté des sciences de Toulouse : Mathématiques Y1 - 2021 A1 - Fabio Cavalletti A1 - Nicola Gigli A1 - Flavia Santarcangelo VL - Ser. 6, 30 UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1679/ ER - TY - ABST T1 - A first-order condition for the independence on p of weak gradients Y1 - 2021 A1 - Nicola Gigli A1 - Francesco Nobili ER - TY - ABST T1 - Monotonicity formulas for harmonic functions in RCD(0,N) spaces Y1 - 2021 A1 - Nicola Gigli A1 - Ivan Yuri Violo AB -

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 `(almost) outer volume cone implies (almost) outer metric cone' theorem.

ER - TY - ABST T1 - Parallel transport on non-collapsed $\mathsfRCD(K,N)$ spaces Y1 - 2021 A1 - Emanuele Caputo A1 - Nicola Gigli A1 - Enrico Pasqualetto AB -

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.

ER - TY - JOUR T1 - Benamou–Brenier and duality formulas for the entropic cost on RCD*(K,N) spaces JF - Probability Theory and Related Fields Y1 - 2019 A1 - Nicola Gigli A1 - Luca Tamanini AB -

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 -