In Several Complex Variables, one of the most celebrated results is due to Charles Fefferman, who proved that any biholomorphic map between two bounded strongly pseudoconvex domains extends to a diffeomorphism up to the boundary. Since then, many generalizations have been obtained under different geometric conditions relying on the Bergman kernel. These approaches, however, require smooth boundary assumptions. In another direction, extensions of biholomorphic and proper holomorphic maps have been studied using the Carathéodory and Kobayashi metrics, which depend on the metric bounds. These metric bounds are also explored to show coarse geometric properties of the invariant metrics and some applications in Several Complex Variables.

Recently, the coarse geometric properties, visibility, and Gromov hyperbolicity have been used to get continuous extensions of Kobayashi isometries. The main advantage of these coarse notions is that one can apply them when the domains are not smooth. Moreover, these techniques enable us to obtain extension results for a general class of maps, quasi-isometries with respect to the invariant metric: the Kobayashi metric, the Bergman metric, the Carathéodory metric, the Kähler–Einstein metric, and so on.

In this talk, we will explore analogous extension phenomena for isometries between complex domains with respect to the invariant metrics. We show that an isometry with respect to an invariant metric between two m-convex domains extends bi-Hölder continuously up to the boundary. In the biholomorphic setting, such a result was established earlier by Mercer, while for isometries, only partial results are known under the assumption of C1, Dini-smooth boundaries. We will also present some other extension results for isometries and quasi-isometries with respect to invariant metrics. This work is under preparation.