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In this paper we introduce a notion of the Gromov-Hausdorff distance withboundary, denoted by $d_{GHB}$, to construct a framework of convergence ofnoncomplete metric spaces. We show that a class of bounded $A$-uniform spaceswith diameter bounded from below is a complete metric space with respect to$d_{GHB}$. As an application we show the stability of Gromov hyperbolicity,roughly starlike property, uniformization, quasihyperbolization, and boundaryof Gromov hyperbolic spaces under appropriate notions of convergence andassumptions.

Gromov-Hausdorff distance with boundary and its application to Gromov hyperbolic spaces and uniform spaces