# Conformal equivalence of visual metrics in pseudoconvex domains

@article{Capogna2017ConformalEO, title={Conformal equivalence of visual metrics in pseudoconvex domains}, author={Luca Capogna and Enrico Le Donne}, journal={Mathematische Annalen}, year={2017}, volume={379}, pages={743-763} }

We refine estimates introduced by Balogh and Bonk, to show that the boundary extensions of isometries between bounded, smooth strongly pseudoconvex domains in $${\mathbb {C}}^n$$ C n are conformal with respect to the sub-Riemannian metric induced by the Levi form. As a corollary we obtain an alternative proof of a result of Fefferman on smooth extensions of biholomorphic mappings between bounded smooth pseudoconvex domains. The proofs are inspired by Mostow’s proof of his rigidity theorem and…

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## References

SHOWING 1-10 OF 41 REFERENCES

### Geometric and analytic quasiconformality in metric measure spaces

- Mathematics
- 2010

We prove the equivalence between geometric and analytic definitions of quasiconformality for a homeomorphism $f\colon X\rightarrow Y$ between arbitrary locally finite separable metric measure spaces,…

### Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvex domains

- Mathematics
- 2000

Abstract. We give an estimate for the distance function related to the Kobayashi metric on a bounded strictly pseudoconvex domain with C2-smooth boundary. Our formula relates the distance function on…

### Gromov hyperbolicity and the Kobayashi metric on convex domains of finite type

- Mathematics
- 2014

In this paper we prove necessary and sufficient conditions for the Kobayashi metric on a convex domain to be Gromov hyperbolic. In particular we show that for convex domains with $$C^\infty $$C∞…

### Conformality and Q-harmonicity in sub-Riemannian manifolds

- MathematicsJournal de Mathématiques Pures et Appliquées
- 2019

### Asymptotic Expansions of Invariant Metrics of Strictly Pseudoconvex Domains

- MathematicsCanadian Mathematical Bulletin
- 1995

Abstract In this paper we obtain the asymptotic expansions of the Carathéodory and Kobayashi metrics of strictly pseudoconvex domains with C∞ smooth boundaries in ℂn. The main result of this paper…

### Embeddings of Gromov hyperbolic spaces

- Mathematics
- 2000

Abstract. It is shown that a Gromov hyperbolic geodesic metric space X with bounded growth at some scale is roughly quasi-isometric to a convex subset of hyperbolic space. If one is allowed to…

### Uniformizing Gromov hyperbolic spaces

- Mathematics
- 2001

— The unit disk in the complex plane has two conformally related lives: one as an incomplete space with the metric inherited from R 2 , the other as a complete Riemannian 2-manifold of constant…

### Proper holomorphic mappings extend smoothly to the boundary

- Mathematics
- 1982

Kohn has proved that the Bergman projection associated to a smooth bounded domain D maps C°°(D) into C°°(D) when D is strictly pseudoconvex [11], and more generally, when the boundary of D satisfies…

### Metric Spaces of Non-Positive Curvature

- Mathematics
- 1999

This book describes the global properties of simply-connected spaces that are non-positively curved in the sense of A. D. Alexandrov, and the structure of groups which act on such spaces by…

### Local boundary regularity of holomorphic mappings

- Mathematics
- 1980

Let f be a holomorphic mapping between two bounded domains D and D' in complex space ℂn. Suppose that D and D' contain smooth real hypersurfaces Γ and Γ′ as open subsets of their respective…