On the differential structure of metric measure spaces and applications

By Nicola Gigli. 2012

428 citations recorded by the source corpus.

The main goals of this paper are: i) To develop an abstract differential calculus on metric measure spaces by investigating the duality relations between differentials and gradients of Sobolev functions. This will be achieved without calling into play any sort of analysis in charts, our assumptions being: the metric space is complete and separable and the measure is Borel, non negative and locally finite. ii) To employ these notions of calculus to provide, via integration by parts, a general definition of distributional Laplacian, thus giving a meaning to an expression like $\Delta g=\mu$, where $g$ is a function and $\mu$ is a measure. iii) To show that on spaces with Ricci curvature bounded from below and dimension bounded from above, the Laplacian of the distance function is always a measure and that this measure has the standard sharp comparison properties. This result requires an additional assumption on the space, which reduces to strict convexity of the norm in the case of smooth Finsler structures and is always satisfied on spaces with linear Laplacian, a situation which is analyzed in detail.

Subfields: Applied Mathematics, Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/7c2f6e70f0fe30de1f76e5c93315adca0670a3c9

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Citations in the corpus

From volume cone to metric cone in the nonsmooth setting (2015, methodology): “11 in [11]) yields ∆b̄ = φ′′ ◦ b|Db|m+ φ′ ◦ b∆b = ( 2bφ′′ ◦ b +Nφ′ ◦ b ) m, Given that 2bφ′′ ◦ b + Nφ′ ◦ b ∈ L∞ ∩ W 1,2(X), this identity shows that b̄ ∈ D(∆) with ∆b̄ ∈ L∞ ∩W 1,2(X), yielding the desired conclusion.”
Euclidean spaces as weak tangents of infinitesimally Hilbertian metric measure spaces with Ricci curvature bounded below (2013, background): “K;N / has been proposed in [2] (see also [1, 4, 18, 19, 28]): it consists in requiring that the space .”
On the Bakry-\'Emery condition, the gradient estimates and the Local-to-Global property of RCD*(K,N) metric measure spaces (2013, neutral): “(c) (X, d,m) is infinitesimally Hilbertian according to the terminology introduced in [21], i.”
Well posedness of Lagrangian flows and continuity equations in metric measure spaces (2014, methodology): “(iii) (X, d,m) is infinitesimally Hilbertian according to the terminology introduced in [Gigli 2012], i.”
Li–Yau and Harnack type inequalities in RCD∗(K,N) metric measure spaces (2013, methodology): “Let us recall that infinitesimal Hilbertianity has proved to be a very useful assumption both from the analytic point of view (see for instance [5], [17] and [21]) and from the geometric one (for instance in [30] the second named author defined a notion of angle in such spaces).”
Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds (2012, background): “(X, d,m) is infinitesimally Hilbertian according to [20]), ECh := 2Ch.”
Convergence of pointed non‐compact metric measure spaces and stability of Ricci curvature bounds and heat flows (2013, background): “There are several, equivalent, definitions of Sobolev functions from a metric measure space (X, d,m) to R: we consider here only the case p = 2 following the approach proposed in [8]; we refer the reader to [2, 9] for the case of a general summability exponent p ∈ [1,∞) and to [26] for further devel”
New formulas for the Laplacian of distance functions and applications (2018, methodology): “The equivalence between RCD(K,N) and Bochner inequality (properly written in a weak form, called BakryÉmery condition BE(K,N)) was proved for N =∞ by Ambrosio-Gigli-Savaré [4, 5], and in the finite dimensional case by Erbar-Kuwada-Sturm [28] and Ambrosio-Mondino-Savaré [6].”