Lempert theorem for strongly linearly convex domains

Abstract

Abstract. In 1984 L. Lempert showed that the Lempert function and the Carathéodory distance coincide on non-planar bounded strongly linearly convex domains with real analytic boundaries. Following this paper, we present a slightly modified and more detailed version of the proof. Moreover, the Lempert Theorem is proved for non-planar bounded C 2 -smooth strongly linearly convex domains.The aim of this paper is to present a detailed version of the proof of the Lempert Theorem in the case of non-planar bounded strongly linearly convex domains with smooth boundaries. The original Lempert's proof is presented only in proceedings of a conference (see [6]) with a very limited access and at some places it was quite sketchy. We were encouraged by some colleagues to prepare an extended version of the proof in which all doubts could be removed and some of details of the proofs could be simplified. We hope to have done it below. Certainly, the idea of the proof belongs entirely to Lempert. The main differences, we would like to draw attention to, are• results are obtained in C 2 -smooth case; • the notion of stationary mappings and E-mappings is separated;• a geometry of domains is investigated only in neighborhoods of boundaries of stationary mappings (viewed as boundaries of analytic discs) -this allows us to obtain localization properties for stationary mappings; • boundary properties of strongly convex domains are expressed in terms of the squares of their Minkowski functionals.Additional motivation for presenting the proof is the fact, showed recently in [7], that the so-called symmetrized bidisc may be exhausted by strongly linearly convex domains. On the other hand it cannot be exhausted by domains biholomorphic to convex ones ([1]). Therefore, the equality of the Lempert function and the Carathéodory distance for strongly linearly convex domains does not follow directly from [5].

Goldilocks domains, a weak notion of visibility, and applications

“…The result has been proved in [Lem84] for domains with real-analytic boundary, but the arguments therein can be adapted to the smooth case; also see [KW13]. We refer the reader to [Lem84] or [KW13] for a definition of strong linear convexity. It follows from the discussion on smoothly-bounded Hartogs domains in [APS04, Chapter 2] that strongly linearly convex domains need not necessarily be convex.…”

Norm preserving extensionsof bounded holomorphic functions

“…There are a neighbourhood U of a and a smooth mapping Φ : U × D → C d such that, for any z near a, a disc Φ(z, ·) is a geodesic passing through a 0 and z such that Φ(z, 0) = a 0 and Φ(z, t z ) = z for some t z > 0. It also follows from Lempert's theorem [18,19] (see [16] for an exposition) that z → t z is smooth. Observe that Φ(a, λ) = f (m t 0 (λ)).…”

Parameter dependence of complex geodesics and its applications

“…We now recall the following intrinsic characterization of complex geodesics on strongly linearly convex domains, which is due to Lempert [Lem84] (see also [KW13] for an exposition) and plays a fundamental role in the rest of this paper.…”