This is a very active field, not least because of the fascinating relations with complex algebraic and arithmetic geometry. Conference on complex geometry, dynamical systems and foliation theory 15 19 may 2017 venue. International conference in nevanlinna theory and complex geometry march 14 18, 2012. This is a set of notes from a 5day doityourself or perhaps discoverityourself introduction to hyperbolic geometry. Euclidean geometry by rich cochrane and andrew mcgettigan. Complex hyperbolicity is a notion in complex geometry which could be understood either from the point of view of value distribution of entire holomorphic curves in a complex manifold, or the point of view of existence of nonpositive curved metric. Hyperbolicity of projective hypersurfaces simone diverio. Hyperbolicity in complex geometry harvard mathematics. A similar relative hyperbolicity result is proved for the mapping class group of a surface. Hyperbolic geometry was created in the rst half of the nineteenth century in the midst of attempts to understand euclids axiomatic basis for geometry. Submitted on 21 apr 1998 v1, last revised 11 aug 1998 this version, v2 abstract. Most of the time we will use the latter interpretation. Another standard model for complex hyperbolic space is a paraboloid in c2 called the siegel domain.
Hyperbolicity of complex varieties these are notes. We also mentioned in the beginning of the course about euclids fifth postulate. In mathematics and especially complex geometry, the kobayashi metric is a pseudometric intrinsically associated to any complex manifold. In a saccheri quadrilateral, the summit is longer than the base andthe segment joiningtheir midpoints is shorter than each arm. The anglesum of a triangle does not exceed two right angles, or 180. We give another approach to nevanlinna theory using di. Hyperbolic geometry is also used to study surface groups. For compact complex manifolds the converse of this result is true br2. It is not hard to see that the complex is finitedimensional, but locally infinite. We discuss the techniques and methods for the hyperbolicity problems for submanifolds and their complements in abelian varieties and the complex projective space.
We assume that hyperbolic geometry underlies these networks, and we show that with this assumption, heterogeneous degree distributions and strong clustering in complex networks emerge naturally as simple reflections of the negative curvature and metric property of the. Thurston introduced tools from hyperbolic geometry to study knots that led to new geometric invariants, especially hyperbolic volume. We show how the global clustering coe cient can be. R 1 and consider a symmetric bilinear form of signature n. Weilpetersson metric and hyperbolicity problems of some. Here the notion of algebraic kobayashi hyperbolicity is as follows. The boundary in complex hyperbolic spaces, known as spherical cr. The hyperbolicity problem in complex geometry studies the conditions for a given complex manifold x to be hyperbolic. Models there are many other models of ndimensional hyperbolic space.
We do a rigorous analysis of clustering and characterize the global clustering coe cient in terms of the parameters of the model. For statements involving the vanishing of, its vanishing in the second interpretation for all choices of the coordinate. In hyperbolic geometry, is negative, so the square root is of a positive number. Pdf hypercomplex hyperbolic geometry sarah markham. Connection between complex hyperbolic and complex trigonometric functions. Math 8 1999 103149 abstract the complex of curves on a surface is a simplicial complex whose vertices are homotopy classes of simple closed curves, and whose simplices are sets of homotopy classes which can be realized disjointly. Basics of hyperbolic geometry rich schwartz october 8, 2007 the purpose of this handout is to explain some of the basics of hyperbolic geometry. This is a great mathematics book cover the following topics. Book of abstracts another approach to nevanlinna theory from di. Intersection numbers and the hyperbolicity of the curve. The rank one symmetric spaces of noncompact type are the real, complex, quaternionic and octonionic hyperbolic spaces.
Then holomorphic discs in xsatisfy an isoperimetric linear inequality, i. Hyperbolicity in complex geometry 3 on c which is the coe. Introduction consider a compact oriented surface s of genus g. These lecture notes are based on a minicourse given. Compacti cation and isometries of hyperbolic space 36 2.
Thoroughly updated, featuring new material on important topics such as hyperbolic geometry in higher dimensions and generalizations of hyperbolicity includes full solutions for all exercises successful first edition sold over 800 copies in north america. Instead, we will develop hyperbolic geometry in a way that emphasises the similarities and more interestingly. Now enters geometry in 1980s, william thurstons seminal work established a strong connection between hyperbolic geometry and knot theory, namely that most knot complements are hyperbolic. Weilpetersson metric and hyperbolicity problems of some families of polarized manifolds conference on complex geometry, dynamic systems and foliation theory institute for mathematical sciceces national university of singapore may 1519, 2017 saikee yeung purdue university may 16, 2017. It is one type of noneuclidean geometry, that is, a geometry that discards one of euclids axioms. Diy hyperbolic geometry kathryn mann written for mathcamp 2015 abstract and guide to the reader. Let points e and f be the midpoints of the base and summit, respectively. Everything from geodesics to gaussbonnet, starting with a. This is analogous to the the half space model of h2 r. They have profound applications to the study of complex variables, to the topology of two and threedimensional manifolds, to the study of nitely presented in nite groups, to physics, and to other disparate elds of mathematics. Request pdf hyperbolicity in complex geometry a complex manifold is said to be hyperbolic if there exists no nonconstant holomorphic map from the affine complex line to it. A complex manifold x is called kobayashi hyperbolic if its kobayashi pseudodistance is nondegenerate. However, while there are a number of books on analysis in such spaces, this.
Hyperbolicity problems have a long history and trace back to the small picard theorem and the hyperbolicity of compact riemann surfaces of genus. The definition, introduced by mikhael gromov, generalizes the metric properties of classical hyperbolic geometry and of tr. It measures the treelikeness of a graph from a metric viewpoint. Complex hyperbolic geometry oxford mathematical monographs. The complex of curves on a surface is a simplicial complex whose vertices are homotopy classes of simple closed curves, and whose simplices are sets of homotopy classes which can be realized disjointly. Further explanation of complex networks and hyperbolicity will be given in the rest of this section. This leads us to the important concept of a hyperbolic manifold or. The two commonly used notions are brody hyperbolicity and kobayashi hyperbolicity. However, while there are a number of books on analysis in such spaces, this book is the first to. In hyperbolic geometry, the circumference of a circle of radius r is greater than. Summer school on algebraic geometry, held at santa cruz in july 1995.
Chapter 1 geometry of real and complex hyperbolic space 1. On hyperbolic geometry structure of complex networks. More precisely, a graph parameter called hyperbolicity, which is related to hyperbolic geometry, is studied in the context of complex networks. Debarre hyperbolicity of complex varieties, debarrecourse2. Many deep problems remain unsolved, some to do with a mysterious connection with arithmetic. Schedule revised 3812 all talks held in 127 hayeshealy hall. Einstein and minkowski found in noneuclidean geometry a. Complex hyperbolic geometry is a particularly rich field, drawing on riemannian geometry, complex analysis, symplectic and contact geometry, lie group theory, and harmonic analysis. The geometry of knot complements city university of new.
Free geometry books download ebooks online textbooks. We discuss the techniques and methods for the hyperbolicity problems for submanifolds and their complements in abelian varieties and the complex projective. Real hyperbolic geometry is widely studied complex hyperbolic geometry less so, whilst quaternionic hyperbolic geometry is still. Mathematics, math research, mathematical modeling, mathematical programming, math articles, applied math.
A complex manifold is said to be hyperbolic if there exists no nonconstant holomorphic map from the affine complex line to it. As complex hyperbolic 1space is just the unit disc in c with the poincar. Hyperbolicity is a largescale property, and is very useful to the study of certain. It was introduced by harvey as an analogy, in the context of. But geometry is concerned about the metric, the way things are measured. See figure 5 in 1 for a schematic of how the various projections are related.
X y be a smooth morphism of complex manifolds, where. Complex hyperbolic geometry 5 a consequence is a characterization of kobayashi hyperbolicity in terms of isoperimetric inequalities. In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations depending quantitatively on a nonnegative real number. Complex structures in algebra, geometry, topology, analysis.
Chapter 15 hyperbolic geometry math 4520, spring 2015 so far we have talked mostly about the incidence structure of points, lines and circles. Free geometry books download ebooks online textbooks tutorials. This book presents recent advances on kobayashi hyperbolicity in complex geometry, especially in connection with projective hypersurfaces. Hyperbolicity is a quasiisometry invariant, from which one can deduce immediately that certain variations on the curve complex are also hyperbolic 1. Incident, parallel, and ultraparallel subspaces 39 2. Clustering and the hyperbolic geometry of complex networks. The arithmetic and the geometry of kobayashi hyperbolicity. On the hyperbolicity of surfaces of general type with small k. It gives generalizations of nevanlinna theory for meromorphic functions on complete k. Elliptic, parabolic, and hyperbolic isometries 38 2. Hyperbolicity problems in higherdimensional complex geometry have been intensively studied in recent years.
We develop a geometric framework to study the structure and function of complex networks. The complex of curves exactly encodes the intersection patterns of this family of regions it is the nerve of the family, and we show that its hyperbolicity means that the teichmuller space is relatively hyperbolic with respect to this family. Hyperbolicity in complex geometry harvard university. The arithmetic and the geometry of kobayashi hyperbolicity 3 figure 1. Complex geometry, dynamical systems and foliation theory ims. Kobayashi hyperbolicity implies brody hyperbolicity. Equilateral triangle, perpendicular bisector, angle bisector, angle made by lines, the regular hexagon, addition and subtraction of lengths, addition and subtraction of angles, perpendicular lines, parallel lines and angles, constructing parallel. Now we study some properties of hyperbolic geometry which do not hold in euclidean geometry. In other words, kobayashi hyperbolicity implies brody hyperbolicity. Iin complex geometry, we can measure negativity, or hyperbolicity, of a k ahler manifold m.
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