Reshetnyak's Theory of Subharmonic Metrics / / François Fillastre and Dmitriy Slutskiy

• Autore: Fillastre François
• Pubblicazione: Cham, Switzerland : , : Springer, Springer Nature Switzerland AG, , [2023]; ©2023
• Edizione: First edition.
• Descrizione fisica: 1 online resource (389 pages)
• Disciplina: 516.2
• Soggetto topico: Curves; Manifolds (Mathematics); Subharmonic functions
• Persona (resp. second.): SlutskiyDmitriy
• Nota di contenuto: Intro -- Foreword -- Preface -- Contents -- 1 How I Got Involved in Research on Two-Dimensional Manifolds of Bounded Curvature -- 2 On Alexandrov's Surfaces with Bounded Integral Curvature -- 2.1 Introduction -- 2.2 Definition of Alexandrov's Surfaces -- 2.3 On the Weyl Problem -- 2.4 Alexandrov Surfaces Obtained by Gluing Riemannian Pieces -- 2.5 Conformal Structure and Uniformization of Smooth Surfaces -- 2.6 Green Kernel and Potential -- 2.7 The Conformal Representation of Alexandrov Surfaces -- 2.8 Toward a Classification of Compact Alexandrov Surfaces -- 2.9 Some Questions and Problems -- References -- 3 Riemannian Surfaces with Simple Singularities -- 3.1 Local Description -- 3.2 Global Description -- 3.3 Some Global Geometry -- 3.4 Classifying Flat Metrics -- 3.5 The Berger-Nirenberg Problem on Surfaces with Divisors -- 3.6 Spherical Polyhedra -- References -- 4 An Introduction to Reshetnyak's Theory of Subharmonic Distances -- 4.1 Introduction -- Terminology -- 4.2 Curves in the Plane -- 4.2.1 Rotation, Turn, and Laplacian -- 4.2.1.1 Rotation of Arcs in the Plane -- 4.2.1.2 Turn of Arcs, Laplacian, and Curvature for Riemannian Metric -- 4.2.1.3 Angle Under Which an Arc Is Seen: Left and Right Turn of Arcs -- 4.2.1.4 Flat Cones and Weak Laplacian -- 4.2.2 Arcs of Bounded Rotation -- 4.2.2.1 Classical Results -- 4.2.2.2 Arcs of Bounded Rotation and Convex Functions -- 4.2.2.3 Bounded Rotation and Bounded Turn -- 4.3 Subharmonic Functions -- 4.3.1 Definition -- 4.3.2 Approximation -- 4.3.3 Polar Sets -- 4.3.4 Delta-Subharmonic Functions -- 4.3.5 Localization of Delta-Subharmonic Functions -- 4.4 Subharmonic Distances on the Plane -- 4.4.1 Length of Arcs -- 4.4.2 Length Distance -- 4.4.3 Canonical Stretching and Points at Infinity -- 4.4.4 Distances Convergence Theorem and Consequences -- 4.4.4.1 Convergence of Distances. ; 4.4.4.2 Consequences on Curves -- 4.4.5 Contraction onto a Cone -- 4.4.6 Two-Dimensional Manifolds of Bounded Curvature -- 4.5 Conformal Aspects of Subharmonic Distances -- 4.5.1 Conformal Mappings -- 4.5.2 A Glimpse to Riemann Surfaces -- 4.6 Other Results -- 4.6.1 Isothermal Coordinates on Two-Dimensional Manifolds of Bounded Curvature -- 4.6.2 Turn and Curvature in a Two-Dimensional Manifold of Bounded Curvature -- 4.6.3 Lipschitz Approximation and the Area Measure -- 4.6.4 Non-Positive Curvature and Isoperimetric Inequality -- References -- 5 Isothermal Coordinates on Manifolds of Bounded Curvature -- References -- 6 Study of Manifolds of Bounded Curvature Using Isothermal Coordinates -- References -- 7 Isothermal Coordinates on Manifolds of Bounded Curvature I -- 1 Isothermal Coordinates in Two-Dimensional Riemannian Manifolds -- 2 Definition of Manifolds of Bounded Curvature -- 3 Curves of Bounded Rotation -- 4 Statements of Results -- 5 Metrics Convergence Theorem -- 6 Reduction of the Proof of the Metrics Convergence Theorem to the Proof of Three Basic Lemmas -- 7 Proof of Theorem I -- 8 Proof of Theorem II -- References -- 8 Isothermal Coordinates on Manifolds of Bounded Curvature II -- 1 First Basic Lemma -- 2 Second Basic Lemma -- 3 Third Basic Lemma -- References -- 9 On Isoperimetric Property of Two-dimensional Manifolds with Curvature Bounded from Above by K -- 1 Main Definitions and Statement of Results -- 2 The λ Regular Case -- 3 Local Boundedness of the Function λ -- 4 Approximation by Riemannian Metrics -- 5 Properties of Functions of Bounded Curvatures -- 6 Proof of the Main Theorem -- References -- 10 On a Special Mapping of a Cone onto a Polyhedron -- 1 Main Definition and Statement of the Results -- 2 Auxiliary Statements -- 3 Proof of Theorem 1 -- 4 Proof of Theorem 2. ; 5 Some Applications of Theorems 1 and 2 -- References -- 11 On a Special Mapping of a Cone in a Manifold of BoundedCurvature -- 1 Main Definitions and Statements of the Results -- 2 Some Statements About Metrics Convergence -- 3 Construction of Polyhedra Converging to a Given Domain -- 4 Correction of the Sequence Dn -- 5 Construction of the Convergence Sequence of Cones -- 6 Proof of Theorem 1 -- 7 Toward the Axiomatic of Manifolds of Bounded Curvature -- References -- 12 Arc Length in Manifolds of Bounded Curvature with an Isothermal Metric -- 1 Definition and Statement of the Result -- 2 Some Properties of the Logarithmic Potential -- 3 Definition of the Integral along a Curve -- 4 Proof of the Main Theorem -- 5 Example of a Non-rectifiable Curve of Finite Length in an (M,λ) -- References -- 13 Turn of Curves in Manifolds of Bounded Curvature with Isothermal Metric -- 1. Introduction -- 2. Angular Function of a Curve, Notion of Side, Rotation of a Curve -- 3. The Formal Turn -- 4. Some Lemmas on Weak Convergence of Measures -- 5. Lemmas on Passage to the Limit for the Formal Turn -- 6. Metric Localization Theorem -- 7. Non-positivity of Turn of Shortest Arcs -- 8. Invariance of the Turn of a Curve -- 9. Curves of Bounded Turn -- References -- 14 On the Potential Theoretic Aspect of Alexandrov Surface Theory -- 1 Introduction -- 2 Auxiliary Results Concerning Subharmonic Functions -- 3 Proof of Theorem A -- 4 A Potential Theoretic Definition of Curve Length -- References -- Index.
• Titolo autorizzato: Reshetnyak's Theory of Subharmonic Metrics
• ISBN: 3-031-24255-6
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910746282503321
• Lo trovi qui: Univ. Federico II