Conformal equivalence of visual metrics in pseudoconvex domains

Luca Capogna,Enrico Le Donne

doi:10.1007/s00208-020-01962-1

Luca Capogna, Enrico Le Donne

Open Access

https://doi.org/10.1007/s00208-020-01962-1

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Abstract

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 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 are based on the asymptotic hyperbolic character of the Kobayashi or Bergman metrics and on the Bonk-Schramm hyperbolic fillings.

Highlights

Let D ⊂ Cn(n ≥ 2) be a bounded, strongly pseudo-convex domain with C∞-smooth boundary
They show that the Carnot–Carathéodory metric dCC corresponding to the Levi form on ∂ D, determines the canonical class of snowflake equivalent visual metrics on ∂ D
Among these we recall that every quasi-isometry between such spaces extends to a quasi-conformal map between the visual boundaries, endowed with their families of visual metrics, see for instance [6,17] and references therein

Summary

Introduction

Let D ⊂ Cn(n ≥ 2) be a bounded, strongly pseudo-convex domain with C∞-smooth boundary. Results from the theory of Gromov hyperbolic spaces can be immediately applied in this setting Among these we recall that every quasi-isometry between such spaces extends to a quasi-conformal map between the visual boundaries, endowed with their families of visual metrics, see for instance [6,17] and references therein. Theorem 1.1 Let D1, D2 ⊂ Cn be bounded strongly pseudoconvex C∞-smooth domains and denote by dK the distance function corresponding to a Finsler structure K satisfying (2.8), and by dCC the Carnot–Carathéodory distance on the boundaries induced by the Levi form. The contribution of the present paper does not lie so much in an innovation on a technical level, but rather in two new insights: namely, that one can deduce Fefferman’s result from the conformality of the boundary extension and that one can prove this with relative ease from a combination of the general theory of Gromov hyperbolic spaces in combination with careful estimates in Theorem 4.1. For more results along this line, see the recent, interesting work of Zimmer in [24]

Distorsion in subRiemannian geometry

Pseudoconvex domains and hermitian metrics

Gromov hyperbolicity

Boundary distances of hyperbolic fillings

Lemmata