Comparison of invariant functions and metrics

Łukasz Kosiński

doi:10.1007/s00013-013-0596-y

Abstract

It is shown that all invariant metrics and functions on a bounded \({\mathcal{C}^2}\) -smooth domain coincide on an open non-empty subset. The existence of Lempert–Burns–Krantz discs in \({\mathcal{C}^2}\) -smooth domains and other possible applications are also discussed.

Highlights

It is well known that if is a contractible family of functions (respectively (δ)G a contractible family of pseudometrics), where G goes through the family of all domains in Cn, cG ≤ dG ≤ lG
The Lempert Theorem may be formulated as follows: on any convex or smooth C-convex domain of Cn, all invariant metrics are equal. This result is surprising as the functions and metrics mentioned above are holomorphic objects and notions of convexity and C-convexity are just algebraic conditions
We show that modifying this method, we may obtain the existence of Lempert–Burns–Krantz discs in C2 smooth domains

Summary

Introduction

The Lempert Theorem may be formulated as follows: on any convex or smooth C-convex domain of Cn, all invariant metrics are equal. If necessary, to a subsequence we may assume that fμ converges to a mapping f0 : D → Bn. Since f0(0) = z ∈ Bn we get that f0(D) ⊂ Bn. the statement of the Lempert Theorem holds on Dμ that is cDν = lDν , we may see that f0 is a complex geodesic in Bn passing through (z, w).

Results

Conclusion