1,721,009 research outputs found
Equivalence of two different notions of tangent bundle on rectifiable metric measure spaces
Get PDFWe prove that for a suitable class of metric measure spaces the abstract notion of tangent module as defined by the first author can be isometrically identified with the space of L2-sections of the ‘Gromov-Hausdorff tangent bundle’. The key assumption that we make is a form of rectifiability for which the space is ‘almost isometrically’ rectifiable (up to m-null sets) via maps that keep under control the reference measure. We point out that RCD∗(K, N) spaces fit in our framework
Partial derivatives in the nonsmooth setting
No full textWe study partial derivatives on the product of two metric measure structures, in particular in connection with calculus via modules as proposed by the first named author in [13]. Our main results are: i) The extension to this non-smooth framework of Schwarz's theorem about symmetry of mixed second derivatives ii) A quite complete set of results relating the property f∈W2,2(X×Y) on one side with that of f(⋅,y)∈W2,2(X) and f(x,⋅)∈W2,2(Y) for a.e. y,x respectively on the other. Here X,Y are RCD spaces so that second order Sobolev spaces are well defined. These results are in turn based upon the study of Sobolev regularity, and of the underlying notion of differential, for a map with values in a Hilbert module: we mainly apply this notion to the map x↦dyf(x,⋅) in order to build, under the appropriate regularity requirements, its differential dxdyf
Riemann curvature tensor on RCD spaces and possible applications
Get PDFWe show that, on every RCD space, it is possible to introduce, by a distributional-like approach, a Riemann curvature tensor. Since, after the works of Petrunin and Zhang–Zhu, we know that finite dimensional Alexandrov spaces are RCD spaces, our construction applies in particular to the Alexandrov setting. We conjecture that an RCD space is Alexandrov if and only if the sectional curvature – defined in terms of such abstract Riemann tensor – is bounded from below
Korevaar–Schoen’s directional energy and Ambrosio’s regular Lagrangian flows
No full textWe develop Korevaar–Schoen’s theory of directional energies for metric-valued Sobolev maps in the case of RCD source spaces; to do so we crucially rely on Ambrosio’s concept of Regular Lagrangian Flow. Our review of Korevaar–Schoen’s spaces brings new (even in the smooth category) insights on some aspects of the theory, in particular concerning the notion of ‘differential of a map along a vector field’ and about the parallelogram identity for CAT(0) targets. To achieve these, one of the ingredients we use is a new (even in the Euclidean setting) stability result for Regular Lagrangian Flows
Benamou–Brenier and duality formulas for the entropic cost on RCD∗(K, N) spaces
Get PDFIn this paper we prove that, within the framework of RCD∗(K, N) spaces with N< ∞, 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 RCD∗(K, N) spaces and our results are new even in this setting
A first-order condition for the independence on p of weak gradients
Get PDFIt is well known that on arbitrary metric measure spaces, the notion of minimal p-weak upper gradient may depend on p. In this paper we investigate how a first-order condition of the metric-measure structure, that we call Bounded Interpolation Property, guarantees that in fact such dependence is not present. We also show that the Bounded Interpolation Property is stable for pointed measure Gromov Hausdorff convergence and holds on a large class of spaces satisfying curvature dimension conditions
Second order differentiation formula on RCD∗(K;N) spaces
Get PDFThe aim of this paper is to prove a second order differentiation formula for H2;2 functions along geodesics in RCD∗(K;N) spaces with K ∈R and N < ∞. This formula is new even in the context of Alexandrov spaces, where second order differentiation is typically related to semiconvexity. We establish this result by showing that W2-geodesics can be approximated up to second order, in a sense which we shall make precise, by entropic interpolations. In turn this is achieved by proving new, even in the smooth setting, estimates concerning entropic interpolations which we believe are interesting on their own. In particular we obtain: • equiboundedness of densities along entropic interpolations, • local equi-Lipschitz continuity of Schrödinger potentials, • uniform weighted L2 control of the Hessian of such potentials. Finally, the techniques adopted in this paper can be used to show that in the RCD setting the viscous solution of the Hamilton-Jacobi equation can be obtained via a vanishing viscosity method, as in the smooth case. With respect to a previous version, where the space was assumed to be compact, in this paper the second order differentiation formula is proved in full generality
Construction of the parallel transport in the Wasserstein space
No full textIn this paper we study the problem of parallel transport in the Wasserstein spaces P_2(R^d). We show that the parallel transport exists along a class of curves whose velocity field is sufficiently smooth, and that we call regular. Furthermore, we show that the class of regular curves is dense in the class of absolutely continuous curves and discuss the problem of parallel transport along geodesics. Most results are extracted from the PhD thesis of the second autho
Monotonicity Formulas for Harmonic Functions in RCD (0 , N) Spaces
Get PDFWe generalize to the RCD (0 , N) setting a family of monotonicity formulas by Colding and Minicozzi for positive harmonic functions in Riemannian manifolds with nonnegative 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 Agostiniani et al. (Invent. Math. 222(3):1033–1101, 2020), 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
Differential of metric valued Sobolev maps
Get PDFWe introduce a notion of differential of a Sobolev map between metric spaces. The differential is given in the framework of tangent and cotangent modules of metric measure spaces, developed by the first author. We prove that our notion is consistent with Kirchheim's metric differential when the source is a Euclidean space, and with the abstract differential provided by the first author when the target is R. We also show compatibility with the concept of co-local weak differential introduced by Convent and Van Schaftingen
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