The Characterizations And Properties Of Strongly Hyperbolic Spaces

The thesis mainly studies the characterizations and properties of strongly hyperbolic spaces.In the first chapter,we introduce the research background and significance of this thesis and the definitions of hyperbolic metric,hyperbolic-type metrics,Gromov hyperbolic metric and strongly hyperbolic metric.In addition,we give out the basic properties and related conclu-sions of strongly hyperbolic spaces.Then we analyze the current status of research on Gromov hyperbolic metric and geometry of metrics.In the second chapter,we show the strong hyperbolicity of ultrametric spaces and construct strongly hyperbolic spaces by the sum of metrics.In particular,we present a product construc-tion on strongly hyperbolic metric spaces.We also construct a metric on Ptolemy space and prove its strong hyperbolicity.Finally,we prove that Ibragimov’s metric strongly hyperbolizes any metric spaces without changing their quasiconformal geometry.In the third chapter,we study the growth of metrics along Euclidean line segments and the starlikeness of metric balls.We also study the growth of Ibragimov’s metric in the upper-half space and the unit ball and the convexity of metric balls.In addition,we give out the bounds of metrics for starlike domains in complex domain.Finally,we present some properties of h-short arcs in strongly hyperbolic spaces.In the fourth chapter,we discuss the properties of metric transforms in strongly hyperbolic spaces.We also describe the features of approximate dilation and logarithm-like metric trans-forms that remain strong hyperbolicity of Euclidean spaces.In particular,we give out some specific examples.In the fifth chapter,we summarize the main contents of this thesis and list several problems for further study.