On The Gromov-Hausdorff Limit Of Metric Spaces

Hausdorff distance and its generalized Gromov-Hausdorf distance are important tools in metric geometry.They also have important applications in other branches of mathematics.In this paper,we will discuss the stability of some metric properties in metric spaces under Gromov-Hausdorff convergence,and give a detailed proof of the stability of some metric properties.The specific contents are as follows:In Chapter 1,we mainly introduces the research background,significance and development of Gromov-Hausdorff distance,and the stability of some invariants of metric space under Gromov-Hausdorf convergence.Finally,the main research contents of this paper are briefly introduced.In Chapter 2,we analyze the stability of a class of metric spaces under Gromov-Hausdorff convergence.Firstly,we review the relevant knowledge of Gromov-Hausdorff distance of metric spaces,and then prove that a class of metric spaces determined by a continuous function f,which defines on metric space of all n×n-matrices m is under the Gromov-Hausdorff convergence.This conclusion can be proved some metric features are stable under Gromov-Hausdorff convergent.In Chapter 3,we analyze the stability theorem in Gromov hyperbolic space.We mainly study the Gromov hyperbolicity of Gromov hyperbolic space.We show that if a sequence of Gromov hyperbolic spaces(X_n,d_n)tend to(X,d)in the sense of Gromov-Hausdorff convergence,then the Gromov hyperbolicity?(X_n)of(X_n,d_n)tend to the Gromov hyperbolicity?(x)of(X,d).In Chapter 4,we analyze the stability theorem of non-positive curvature space under Gromov-Hausdorf limit and prove that if a sequence of non-positive curvature metric spaces(X_n,d_n)tend to(X,d)in the sense of Gromov-Hausdorff convergence,then(X_n,d_n)is also a non-positive curvature metric space.In Chapter 5,we analyze the stability of non-quasi-isometry invariants of metric space under Gromov-Hausdorff convergence.The Assouad-Nagata dimension of metric space was considered,and the corresponding theorem was obtained.