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1. Asymptotic shapes for stationary first passage percolation on virtually nilpotent groups

   https://doi.org/10.48550/arXiv.2110.00055  

   Antonio Auffinger; Christian Gorski (September 2021, ArXivorg)

   We study first passage percolation (FPP) with stationary edge weights on Cayley graphs of finitely generated virtually nilpotent groups. Previous works of Benjamini-Tessera and Cantrell-Furman show that scaling limits of such FPP are given by Carnot-Carathéodory metrics on the associated graded nilpotent Lie group. We show a converse, i.e. that for any Cayley graph of a finitely generated nilpotent group, any Carnot-Carathéodory metric on the associated graded nilpotent Lie group is the scaling limit of some FPP with stationary edge weights on that graph. Moreover, for any Cayley graph of any finitely generated virtually nilpotent group, any conjugation-invariant metric is the scaling limit of some FPP with stationary edge weights on that graph. We also show that the conjugation-invariant condition is also a necessary condition in all cases where scaling limits are known to exist. 

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2. Coordinates Adapted to Vector Fields: Canonical Coordinates

   Stovall, Betsy; Street, Brian (December 2018, Geometric and functional analysis)

   Given a finite collection of C1 vector fields on aC2 manifold which span the tangent space at every point, we consider the question of when there is locally a coordinate system in which these vector fields have a higher level of smoothness. For example, when is there a coordinate system in which the vector fields are smooth, or real analytic, or have Zygmund regularity of some finite order? We address this question in a quantitative way, which strengthens and generalizes previous works on the quantitative theory of sub-Riemannian (aka Carnot–Carathéodory) geometry due to Nagel, Stein, and Wainger, Tao and Wright, the second author, and others. Furthermore, we provide a diffeomorphism invariant version of these theories. This is the first part in a three part series of papers. In this paper, we study a particular coordinate system adapted to a collection of vector fields (sometimes called canonical coordinates) and present results related to the above questions which are not quite sharp; these results form the backbone of the series. The methods of this paper are based on techniques from ODEs. In the second paper, we use additional methods from PDEs to obtain the sharp results. In the third paper, we prove results concerning real analyticity and use methods from ODEs. 

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3. Conformal equivalence of visual metrics in pseudoconvex domains

   https://doi.org/10.1007/s00208-020-01962-1  

   Capogna, Luca; Le Donne, Enrico (January 2021, Mathematische Annalen)

   null (Ed.)

   We refine estimates introduced by Balogh and Bonk, to show that the boundary exten- sions of isometries between bounded, smooth strongly pseudoconvex domains in Cn are conformal with respect to the sub-Riemannian metric induced by the Levi form. As a corollary we obtain an alternative proof of a result of Fefferman on smooth exten- sions of biholomorphic mappings between bounded smooth pseudoconvex domains. The proofs are inspired by Mostow’s proof of his rigidity theorem and are based on the asymptotic hyperbolic character of the Kobayashi or Bergman metrics and on the Bonk-Schramm hyperbolic fillings. 

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4. The asymptotic p-Poisson equation as $$p \rightarrow \infty $$ in Carnot-Carathéodory spaces

   https://doi.org/10.1007/s00208-024-02805-z  

   Capogna, Luca; Giovannardi, Gianmarco; Pinamonti, Andrea; Verzellesi, Simone (January 2024, Mathematische Annalen)

   In this paper we study the asymptotic behavior of solutions to the subelliptic p-Poisson equation as $$p\to \infty$$ in Carnot-Carathéodory spaces. In particular, introducing a suitable notion of differentiability, extend the celebrated result of Bhattacharya et al. (Rend Sem Mat Univ Politec Torino Fascicolo Speciale 47:15–68, 1989) and we prove that limits of such solutions solve in the sense of viscosity a hybrid first and second order PDE involving the infinity- Laplacian and the Eikonal equation. 

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5. Carathéodory's metrics on Teichmüller spaces and L-shaped pillowcases. Duke Math. J. 167 (2018), no. 3, 497–535.

   Makovic, Vladimir (July 2018, Duke Math. J)

   One of the most important results in Teichmüller theory is Royden’s theorem, which says that the Teichmüller and Kobayashi metrics agree on the Teichmüller space of a given closed Riemann surface. The problem that remained open is whether the Carathéodory metric agrees with the Teichmüller metric as well. In this article, we prove that these two metrics disagree on each Teichmüller space of a closed surface of genus g≥2. The main step is to establish a criterion to decide when the Teichmüller and Carathéodory metrics agree on the Teichmüller disk corresponding to a rational Jenkins–Strebel differential. 

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