Corpus ID: 54058138


  author={E. L. STOUTi},
In §2, we use this theorem to establish an analogous result in the setting of finite open Riemann surfaces. §§3 and 4 consider certain questions which arise naturally in the course of the proof of this generalization. We mention that the chief result of §2, Theorem 2.6, has been obtained independently by N. L. Ailing [3] who has used methods more highly algebraic than ours. The second matter we shall be concerned with is that of interpolation. If R is a Riemann surface and if £ is a subset of R… Expand
Projections in the Space H ∞ and the Corona Theorem for Coverings of Bordered Riemann Surfaces
Let M be a non-compact connected Riemann surface of finite type, and R ⊂⊂ M be a relatively compact domain such that H1(M,Z) = H1(R,Z). Let R̃ −→ R be a covering. We study the algebra H∞(U) ofExpand
The parametric h-principle for minimal surfaces in Rⁿ and null curves in Cⁿ
Let M be an open Riemann surface. It was proved by Alarcón and Forstnerič [1] that every conformal minimal immersion M → R is isotopic to the real part of a holomorphic null curve M → C. In thisExpand
In this paper we survey recent developments in the classical theory of minimal surfaces in Euclidean spaces which have been obtained as applications of both classical and modern complex analyticExpand
The Oka principle for holomorphic Legendrian curves in C 2 n + 1 Franc Forstnerič and
Let M be a connected open Riemann surface. We prove that the space L (M,C) of all holomorphic Legendrian immersions of M to C, n ≥ 1, endowed with the standard holomorphic contact structure, isExpand
Sufficient conditions for the projective freeness of Banach algebras
Let R be a unital semi-simple commutative complex Banach algebra, and let M(R) denote its maximal ideal space, equipped with the Gelfand topology. Sufficient topological conditions are given on M(R)Expand
Minimal surfaces in Euclidean spaces by way of complex analysis
This is an extended version of my plenary lecture at the 8th European Congress of Mathematics in Portorož on 23 June 2021. The main part of the paper is a survey of recent applications ofExpand
The Calabi–Yau problem for Riemann surfaces with finite genus and countably many ends
In this paper, we show that if $R$ is a compact Riemann surface and $M=R\setminus\,\bigcup_i D_i$ is a domain in $R$ whose complement is a union of countably many pairwise disjoint smoothly boundedExpand
A strong parametric h-principle for complete minimal surfaces
We prove a parametric h-principle for complete nonflat conformal minimal immersions of an open Riemann surface M into Rn, n ≥ 3. It follows that the inclusion of the space of such immersions into theExpand
Holomorphic Legendrian curves in projectivised cotangent bundles
We study holomorphic Legendrian curves in the standard complex contact structure on the projectivised cotangent bundle $X=\mathbb P(T^*Z)$ of a complex manifold $Z$ of dimension at least $2$. WeExpand
Proper superminimal surfaces of given conformal types in the hyperbolic four-space
Let $H^4$ denote the hyperbolic four-space. Given a bordered Riemann surface, $M$, we prove that every smooth conformal superminimal immersion $\overline M\to H^4$ can be approximated uniformly onExpand


Two theorems concerning functions holomorphic on multiply connected domains
1. Let £2 be a finitely connected plane domain whose boundary, d£2, consists of the circles To, I \ , • • • , Tw. We assume Ty lies in the interior of T0 for j= 1, 2, • • • , n. Let A0 be theExpand
A proof of the Corona conjecture for finite open Riemann surfaces
By a finite open Riemann surface is meant a proper, open, connected subset of a compact Riemann surface W whose boundary T is also the boundary of W— X and consists of a finite number of closedExpand
An Interpolation Problem for Bounded Analytic Functions
be possible for a given sequence of points {zv}, I z, 1, and an analytic function f(z) in I z i < 1, I f(z) I _ 1. The result is, however, very implicit ancd gives in a concrete situation very littleExpand
Introduction to Riemann Surfaces
Introduction: 1-1 Algebraic functions and Riemann surfaces 1-2 Plane fluid flows 1-3 Fluid flows on surfaces 1-4 Regular potentials 1-5 Meromorphic functions 1-6 Function theory on a torus GeneralExpand
Open Riemann surfaces and extremal problems on compact subregions
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