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Academic literature on the topic 'Carathéodory metric'

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Published: 6 September 2023

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Journal articles on the topic "Carathéodory metric"

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Fornæss, John Erik, and Lina Lee. "Kobayashi, Carathéodory and Sibony metric." Complex Variables and Elliptic Equations 54, no. 3-4 (March 2009): 293–301. http://dx.doi.org/10.1080/17476930902760450.

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Abate, Marco, and Jean-Pierre Vigué. "Isometries for the Carathéodory metric." Proceedings of the American Mathematical Society 136, no. 11 (May 20, 2008): 3905–9. http://dx.doi.org/10.1090/s0002-9939-08-09391-x.

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Ge, Zhong. "Collapsing Riemannian Metrics to Carnot-Caratheodory Metrics and Laplacians to Sub-Laplacians." Canadian Journal of Mathematics 45, no. 3 (June 1, 1993): 537–53. http://dx.doi.org/10.4153/cjm-1993-028-6.

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AbstractWe study the asymptotic behavior of the Laplacian on functions when the underlying Riemannian metric is collapsed to a Carnot-Carathéodory metric. We obtain a uniform short time asymptotics for the trace of the heat kernel in the case when the limit Carnot-Carathéodory metric is almost Heisenberg, the limit of which is the result of Beal-Greiner-Stanton, and Stanton-Tartakoff.
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CONNELL, CHRIS, THANG NGUYEN, and RALF SPATZIER. "Carnot metrics, dynamics and local rigidity." Ergodic Theory and Dynamical Systems 42, no. 2 (December 9, 2021): 614–64. http://dx.doi.org/10.1017/etds.2021.116.

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AbstractThis paper develops new techniques for studying smooth dynamical systems in the presence of a Carnot–Carathéodory metric. Principally, we employ the theory of Margulis and Mostow, Métivier, Mitchell, and Pansu on tangent cones to establish resonances between Lyapunov exponents. We apply these results in three different settings. First, we explore rigidity properties of smooth dominated splittings for Anosov diffeomorphisms and flows via associated smooth Carnot–Carathéodory metrics. Second, we obtain local rigidity properties of higher hyperbolic rank metrics in a neighborhood of a locally symmetric one. For the latter application we also prove structural stability of the Brin–Pesin asymptotic holonomy group for frame flows. Finally, we obtain local rigidity properties for uniform lattice actions on the ideal boundary of quaternionic and octonionic symmetric spaces.
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Fu, Siqi. "Asymptotic Expansions of Invariant Metrics of Strictly Pseudoconvex Domains." Canadian Mathematical Bulletin 38, no. 2 (June 1, 1995): 196–206. http://dx.doi.org/10.4153/cmb-1995-028-9.

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AbstractIn this paper we obtain the asymptotic expansions of the Carathéodory and Kobayashi metrics of strictly pseudoconvex domains with C∞ smooth boundaries in ℂn. The main result of this paper can be stated as following:Main Theorem. Let Ω be a strictly pseudoconvex domain with C∞ smooth boundary. Let FΩ(z,X) be either the Carathéodory or the Kobayashi metric of Ω. Let δ(z) be the signed distance from z to ∂Ω with δ(z) < 0 for z ∊ Ω and δ(z) ≥ 0 for z ∉ Ω. Then there exist a neighborhood U of ∂Ω, a constant C > 0, and a continuous function C(z,X):(U ∩ Ω) × ℂn -> ℝ such that and|C(z,X)| ≤ C|X| for z ∊ U ∩ Ω and X ∊ ℂn
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Krushkal, Samuel. "On the Carathéodory metric of universal Teichmüller space." Ukrainian Mathematical Bulletin 19, no. 1 (January 28, 2022): 75–87. http://dx.doi.org/10.37069/1810-3200-2029-19-1-5.

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In contrast to finite dimensional Teichmuller spaces, all non-expanding invariant metrics on the universal Teichmuller space coincide. This important fact found various applications. We give its new, simplified proof based on some deep features of the Grunsky operator, which intrinsically relate to the universal Teichmuller space. This approach also yields a quantitative answer to Ahlfors' question.
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Krushkal, Samuel L. "On the Carathéodory metric of universal Teichmüller space." Journal of Mathematical Sciences 262, no. 2 (April 2022): 184–93. http://dx.doi.org/10.1007/s10958-022-05809-9.

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Selivanova, Svetlana. "Metric Geometry of Nonregular Weighted Carnot–Carathéodory Spaces." Journal of Dynamical and Control Systems 20, no. 1 (December 17, 2013): 123–48. http://dx.doi.org/10.1007/s10883-013-9206-3.

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Nikolov, N. "Continuity and boundary behavior of the Carathéodory metric." Mathematical Notes 67, no. 2 (February 2000): 183–91. http://dx.doi.org/10.1007/bf02686245.

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Świątkowski, Jacek. "Compact 3-manifolds with a flat Carnot-Carathéodory metric." Colloquium Mathematicum 63, no. 1 (1992): 89–105. http://dx.doi.org/10.4064/cm-63-1-89-105.

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Dissertations / Theses on the topic "Carathéodory metric"

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Don, Sebastiano. "Functions of bounded variation in Carnot-Carathéodory spaces." Doctoral thesis, Università degli studi di Padova, 2019. http://hdl.handle.net/11577/3426813.

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We study properties of functions with bounded variation in Carnot-Carathéodory spaces. In Chapter 2 we prove their almost everywhere approximate differentiability and we examine their approximate discontinuity set and the decomposition of their distributional derivatives. Under an additional assumption on the space, called property R, we show that almost all approximate discontinuities are of jump type and we study a representation formula for the jump part of the derivative. In Chapter 3 we prove a rank-one theorem à la G. Alberti for the derivatives of vector-valued maps with bounded variation in a class of Carnot groups that includes all Heisenberg groups H^n with n ≥ 2. Some important tools for the proof are properties linking the horizontal derivatives of a real-valued function with bounded variation to its subgraph. In Chapter 4 we prove a compactness result for bounded sequences (u_j) of functions with bounded variation in metric spaces (X, d_j) where the space X is fixed, but the metric may vary with j. We also provide an application to Carnot-Carathéodory spaces. The results of Chapter 4 are fundamental for the proofs of some facts of Chapter 2.
Analizziamo alcune proprietà di funzioni a variazione limitata in spazi di Carnot-Carathéodory. Nel Capitolo 2 dimostriamo che esse sono approssimativamente differenziabili quasi ovunque, esaminiamo il loro insieme di discontinuità approssimata e la decomposizione della loro derivata distribuzionale. Assumendo un'ipotesi addizionale sullo spazio, che chiamiamo proprietà R, mostriamo che quasi tutti i punti di discontinuità approssimata sono di salto e studiamo una formula per la parte di salto della derivata. Nel Capitolo 3 dimostriamo un teorema di rango uno à la G. Alberti per la derivata distribuzionale di funzioni vettoriali a variazione limitata in una classe di gruppi di Carnot che contiene tutti i gruppi di Heisenberg H^n con n ≥ 2. Uno strumento chiave nella dimostrazione è costituito da alcune proprietà che legano le derivate orizzontali di una funzione a variazione limitata con il suo sottografico. Nel Capitolo 4 dimostriamo un risultato di compattezza per succesioni (u_j) equi-limitate in spazi metrici (X, d_j) quando lo spazio X è fissato ma la metrica può variare con j. Mostriamo inoltre un'applicazione agli spazi di Carnot-Carathéodory. I risultati del Capitolo 4 sono fondamentali per la dimostrazione di alcuni fatti contenuti nel Capitolo 2.
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Lieder, Marc [Verfasser]. "Das Randverhalten der Kobayashi- und Carathéodory-Metrik auf lineal konvexen Gebieten endlichen Typs / vorgelegt von Marc Lieder." 2005. http://d-nb.info/977948994/34.

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Books on the topic "Carathéodory metric"

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Street, Brian. The Calder´on-Zygmund Theory II: Maximal Hypoellipticity. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691162515.003.0002.

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This chapter remains in the single-parameter case and turns to the case when the metric is a Carnot–Carathéodory (or sub-Riemannian) metric. It defines a class of singular integral operators adapted to this metric. The chapter has two major themes. The first is a more general reprise of the trichotomy described in Chapter 1 (Theorem 2.0.29). The second theme is a generalization of the fact that Euclidean singular integral operators are closely related to elliptic partial differential equations. The chapter also introduces a quantitative version of the classical Frobenius theorem from differential geometry. This “quantitative Frobenius theorem” can be thought of as yielding “scaling maps” which are well adapted to the Carnot–Carathéodory geometry, and is of central use throughout the rest of the monograph.
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Book chapters on the topic "Carathéodory metric"

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"II The Carathéodory pseudodistance and the Carathéodory-Reiffen pseudometric." In Invariant Distances and Metrics in Complex Analysis. Berlin, New York: DE GRUYTER, 1993. http://dx.doi.org/10.1515/9783110870312.15.

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