Generalized Ahlfors functions

Abstract

Abstract. Let Σ be a bordered Riemann surface with genus g and m boundary components. Let {γ z } z∈∂Σ be a smooth family of smooth Jordan curves in C which all contain the point 0 in their interior. Let p ∈ Σ and let F be the family of all bounded holomorphic functions f on Σ such that f (p) ≥ 0 and f (z) ∈ γ z for almost every z ∈ ∂Σ. Then there exists a smooth up to the boundary holomorphic function f 0 ∈ F with at most 2g + m − 1 zeros on Σ so that f 0 (z) ∈ γ z for every z ∈ ∂Σ and such that f 0 (p) ≥ f (p) for every f ∈ F. If, in addition, all the curves {γ z } z∈∂Σ are strictly convex, then f 0 is unique among all the functions from the family F .

Identification of a Connection from Cauchy Data on a Riemann Surface with Boundary

“…Indeed, let {φ 1 , .., φ n } be a set of real valued C 3 harmonic functions in M 0 such that M j (φ k ) = δ jk . Such harmonic functions exist due to equation (3.1) of [4] which states that the period matrix is invertible. Then given a real valued harmonic function φ, we consider small perturbations φ ′ := φ + n j=1 ǫ j φ j of φ.…”

Embedding Bordered Riemann Surfaces in Strongly Pseudoconvex Domains

“…In 2004, it was shown by Černe [5,Corollary 1.4] that every finitely bordered planar domain with smooth boundary embeds properly holomorphically and smoothly up to the boundary in any smoothly bounded convex domain in C n for n ≥ 2. The technique developed in his paper (see also the sequel [6] by Černe and Flores) is the basis for our construction in the present paper. It uses solutions to certain Riemann-Hilbert boundary value problems on M , similarly to what was done by the author in [23] and used in [28] in the special case when M is the disc ∆.…”

Embedding bordered Riemann surfaces in strongly pseudoconvex domains

“…In 2004 it was shown by Černe [5,Corollary 1.4] that every finitely bordered planar domain with smooth boundary embeds properly holomorphically and smoothly up to the boundary in any smoothly bounded convex domain in C n for n ≥ 2. The technique developed in his paper (see also the sequel [6] by Černe and Flores) is the basis for our construction in the present paper. It uses solutions to certain Riemann-Hilbert boundary value problems on M , similarly to what was done by the author in [21] and used in [26] in the special case when M is the disc ∆.…”