Oliver Roth

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Golusin-type inequalities for normalized univalent functions are combined with elementary monotonicity arguments to give quick and simple proofs for numerous sharp two-point distortion theorems for conformal maps from the unit disk into (i) the complex plane equipped with euclidean geometry, (ii) the unit disk equipped with hyperbolic geometry, and (iii)(More)
We prove a generalization of the Schwarz–Carathéodory reflection principle for analytic maps f from the unit disk into arbitrary Riemann surfaces equipped with a complete real analytic conformal Riemannian metric λ(w) |dw|. This yields a necessary and sufficient condition for f to have an analytic continuation in terms of the pullback of the metric λ(w)(More)
Abstract. A classical result of Nitsche [21] about the behaviour of the solutions to the Liouville equation ∆u = 4 e2u near isolated singularities is generalized to solutions of the Gaussian curvature equation ∆u = −κ(z) e2u where κ is a negative Hölder continuous function. As an application a higher–order version of the Yau–Ahlfors–Schwarz lemma for(More)
Conformal metrics connect complex analysis, differential geometry and partial differential equations. They were used by Schwarz [42], Poincaré [38], Picard [35, 36, 37] and Bieberbach [4, 5], but it was recognized by Ahlfors [3] and Heins [13] that they are ubiquitous in complex analysis and geometric function theory. They have been instrumental in(More)
genau so i n t e n s i v wie bet de r S toBthe rap i e erfolgte, k 6 n n t e m a n a n n e h m e n , d a b die Spasmoph i l i e al ler W a h r s c h e i n l i c h ke i t n a c h mi' t h 6 h e r e n T r o p f e n g a b e n v o n D~, e twa 3rea l 6 g t t . t/~glich yon o,2 m g in i ccm L 6s ungs mi t t e l , gehe i l t w e r d e n k a n n . Z u m SchluB war n(More)
We establish an extension of Liouville’s classical representation theorem for solutions of the partial differential equation ∆u = 4 e2u and combine this result with methods from nonlinear elliptic PDE to construct holomorphic maps with prescribed critical points and specified boundary behaviour. For instance, we show that for every Blaschke sequence {zj} in(More)
Remark 1.2 (a) For k(z) ≡ −4 the PDE (1.1) reduces to the Liouville equation ∆u = −4 e2u. In this case Theorem 1.1 was proved by Chou and Wan [4, 5] using complex analysis and Liouville’s classical representation formula [12] for the solutions to ∆u = −4 e2u. In the variable curvature case some (nonsharp) estimates for the solutions u of (1.1) satisfying(More)
Abstract. The subject of this paper is Beurling’s celebrated extension of the Riemann mapping theorem [5]. Our point of departure is the observation that the only known proof of the Beurling–Riemann mapping theorem contains a number of gaps which seem inherent in Beurling’s geometric and approximative approach. We provide a complete proof of the(More)