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Geometry --- Geometry, Hyperbolic. --- Géométrie hyperbolique. --- Hyperbolic spaces. --- Espaces hyperboliques. --- Spectral theory (Mathematics) --- Théorie spectrale (mathématiques) --- Asymptotic expansions. --- Développements asymptotiques. --- Asymptotic expansions --- Geometry, Hyperbolic --- Hyperbolic spaces --- Functional analysis --- Hilbert space --- Measure theory --- Transformations (Mathematics) --- Hyperbolic complex manifolds --- Manifolds, Hyperbolic complex --- Spaces, Hyperbolic --- Geometry, Non-Euclidean --- Hyperbolic geometry --- Lobachevski geometry --- Lobatschevski geometry --- Asymptotic developments --- Asymptotes --- Convergence --- Difference equations --- Divergent series --- Functions --- Numerical analysis

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This book presents recent advances on Kobayashi hyperbolicity in complex geometry, especially in connection with projective hypersurfaces. This is a very active field, not least because of the fascinating relations with complex algebraic and arithmetic geometry. Foundational works of Serge Lang and Paul A. Vojta, among others, resulted in precise conjectures regarding the interplay of these research fields (e.g. existence of Zariski dense entire curves should correspond to the (potential) density of rational points). Perhaps one of the conjectures which generated most activity in Kobayashi hyperbolicity theory is the one formed by Kobayashi himself in 1970 which predicts that a very general projective hypersurface of degree large enough does not contain any (non-constant) entire curves. Since the seminal work of Green and Griffiths in 1979, later refined by J.-P. Demailly, J. Noguchi, Y.-T. Siu and others, it became clear that a possible general strategy to attack this problem was to look at particular algebraic differential equations (jet differentials) that every entire curve must satisfy. This has led to some several spectacular results. Describing the state of the art around this conjecture is the main goal of this work.

Mathematics. --- Algebraic geometry. --- Functions of complex variables. --- Differential geometry. --- Differential Geometry. --- Algebraic Geometry. --- Several Complex Variables and Analytic Spaces. --- Hyperbolic spaces. --- Hypersurfaces. --- Hyperbolic complex manifolds --- Manifolds, Hyperbolic complex --- Spaces, Hyperbolic --- Hyperspace --- Surfaces --- Geometry, Non-Euclidean --- Global differential geometry. --- Geometry, algebraic. --- Differential equations, partial. --- Partial differential equations --- Algebraic geometry --- Geometry --- Geometry, Differential --- Complex variables --- Elliptic functions --- Functions of real variables --- Differential geometry

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Presents reissued articles from two classic sources on hyperbolic manifolds. Part I is an exposition of Chapters 8 and 9 of Thurston's pioneering Princeton Notes; there is a new introduction describing recent advances, with an up-to-date bibliography, giving a contemporary context in which the work can be set. Part II expounds the theory of convex hull boundaries and their bending laminations. A new appendix describes recent work. Part III is Thurston's famous paper that presents the notion of earthquakes in hyperbolic geometry and proves the earthquake theorem. The final part introduces the theory of measures on the limit set, drawing attention to related ergodic theory and the exponent of convergence. The book will be welcomed by graduate students and professional mathematicians who want a rigorous introduction to some basic tools essential for the modern theory of hyperbolic manifolds.

Geometry, Hyperbolic --- Hyperbolic spaces --- Three-manifolds (Topology) --- Kleinian groups --- Groups, Kleinian --- Discontinuous groups --- 3-manifolds (Topology) --- Manifolds, Three dimensional (Topology) --- Three-dimensional manifolds (Topology) --- Low-dimensional topology --- Topological manifolds --- Hyperbolic complex manifolds --- Manifolds, Hyperbolic complex --- Spaces, Hyperbolic --- Geometry, Non-Euclidean --- Hyperbolic geometry --- Lobachevski geometry --- Lobatschevski geometry

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This book is an exposition of the theoretical foundations of hyperbolic manifolds. It is intended to be used both as a textbook and as a reference. The book is divided into three parts. The first part is concerned with hyperbolic geometry and discrete groups. The main results are the characterization of hyperbolic reflection groups and Euclidean crystallographic groups. The second part is devoted to the theory of hyperbolic manifolds. The main results are Mostow’s rigidity theorem and the determination of the global geometry of hyperbolic manifolds of finite volume. The third part integrates the first two parts in a development of the theory of hyperbolic orbifolds. The main result is Poincare«s fundamental polyhedron theorem. The exposition if at the level of a second year graduate student with particular emphasis placed on readability and completeness of argument. After reading this book, the reader will have the necessary background to study the current research on hyperbolic manifolds. The second edition is a thorough revision of the first edition that embodies hundreds of changes, corrections, and additions, including over sixty new lemmas, theorems, and corollaries. The new main results are Schl¬afli’s differential formula and the $n$-dimensional Gauss-Bonnet theorem. John G. Ratcliffe is a Professor of Mathematics at Vanderbilt University.

Geometry, Hyperbolic. --- Hyperbolic spaces. --- Hyperbolic complex manifolds --- Manifolds, Hyperbolic complex --- Spaces, Hyperbolic --- Geometry, Non-Euclidean --- Hyperbolic geometry --- Lobachevski geometry --- Lobatschevski geometry --- Geometry, Hyperbolic --- Hyperbolic spaces --- 514.1 --- 514 --- 514.1 General geometry --- General geometry --- 514 Geometry --- Geometry --- Geometry. --- Topology. --- Geometry, algebraic. --- Algebraic Geometry. --- Algebraic geometry --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Mathematics --- Euclid's Elements --- Algebraic geometry. --- Geometry, Algebraic.

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514.13 --- Functions of several complex variables --- Geometry, Differential --- Hyperbolic spaces --- Nevanlinna theory --- Functions, Meromorphic --- Value distribution theory --- Hyperbolic complex manifolds --- Manifolds, Hyperbolic complex --- Spaces, Hyperbolic --- Geometry, Non-Euclidean --- Differential geometry --- Complex variables --- Several complex variables, Functions of --- Functions of complex variables --- Non-Euclidean metric geometries. Lobachevsky geometry. Other hyperbolic geometries. Elliptic geometries --- 514.13 Non-Euclidean metric geometries. Lobachevsky geometry. Other hyperbolic geometries. Elliptic geometries