Nonsmooth differential geometry - An approach tailored for spaces with Ricci curvature bounded from below
By Nicola Gigli. 2014
278 citations recorded by the source corpus.
We discuss in which sense general metric measure spaces possess a first order differential structure. Building on this, we then see that on spaces with Ricci curvature bounded from below a second order calculus can be developed, permitting to define Hessian, covariant/exterior derivatives and Ricci curvature.
Subfields: Applied Mathematics
View Paper: https://www.semanticscholar.org/paper/202dd8fd1dae7bf67777329e18d0227d79cd5147
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In Applied Mathematics
- Elia Brué
- Daniele Semola
- Dario Trevisan
- Nicola Gigli
- Enrico Pasqualetto
- Luigi Ambrosio
- Semola Daniele
- Filippo Santambrogio
Citations in the corpus
- Behaviour of the reference measure on $\sf RCD$ spaces under charts (2016, background): “All these definitions and properties can be found in [9].”
- Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows (2018, methodology): “Following [G18] we introduce the space of “test” functions Test(X, d,m) by Test(X, d,m) := {f ∈ D(∆) ∩ L∞(X,m) : |∇f | ∈ L∞(X) and ∆f ∈ W 1,2(X, d,m)}.”
- Rectifiability of the reduced boundary for sets of finite perimeter over $\RCD(K,N)$ spaces. (2019, methodology): “Gigli in [27] has developed a second order calculus for RCD(K,∞) metric measure spaces.”
- Lectures on Nonsmooth Differential Geometry (2020, background): “The most recent research-level material contained here comes, to a big extent, from the paper [18], see also the survey [20].”
- Well posedness of Lagrangian flows and continuity equations in metric measure spaces (2014, methodology): “Indeed, the BV case seems difficult to reach in the abstract setting, due to the present lack of a covariant derivative (but see [Gigli 2014]).”