Oliver Roth

Pontryagin's Maximum Principle for the Loewner Equation in Higher Dimensions

In this paper we develop a variational method for the Loewner equation in higher dimensions. As a result we obtain a version of Pontryagins maximum principle from optimal control theory for the Loewner equation in several complex variables. Based on recent work of Arosio, Bracci and Wold we then apply our version of the Pontryagin maximum principle to obtain first--order necessary conditions for the extremal functions for a wide class of extremal problems over the set of normalized biholomorphic mappings on the unit ball in

The behaviour of solutions of the Gaussian curvature equation near an isolated boundary point

\mathbb{C}^n

On support points of univalent functions and a disproof of a conjecture of Bombieri

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Pontryagin's maximum principle in geometric function theory

A classical result of Nitsche [22] about the behaviour of the solutions to the Liouville equation ?u = 4e2u near isolated singularities is generalized to solutions of the Gaussian curvature equation ?u = -?(z)e2u where ? is a negative Holder continuous function. As an application a higher-order version of the Yau-Ahlfors-Schwarz lemma for complete conformal Riemannian metrics is obtained.

Critical points of inner functions, nonlinear partial differential equations, and an extension of Liouville's theorem

We consider the linear functional Re(a3 + Aa2) for A C ilR on the set of normalized univalent functions in the unit disk and use the result to disprove a conjecture of Bombieri.

On the local extremum property of the Koebe function

We discuss an infinite-dimensional version of Pontryagins maximum principle as a unified variational method in many familiar classes of analytic functions, and its interrelation with classical variational methods in geometric function theory.

Support Points and the Bieberbach Conjecture in Higher Dimension

AbstractWe establish an extension of Liouvilles classical representation theorem for solutions of the partial differential equation (PDE) Δ u=4 e 2u 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 {z j } in the unit disk there is always a Blaschke product with {z j } as its set of critical points. Our work is closely related to the Berger--Nirenberg problem in differential geometry.

Critical Points, the Gauss Curvature Equation and Blaschke Products

We discuss and compare several necessary criteria for the local extremality of the Koebe mapping in extremal problems for univalent functions. These criteria are applied to study Robertson type inequalities and also to investigate a conjecture of Bombieri.

The Schramm-Loewner equation for multiple slits

We describe some open questions related to support points in the class S0 and introduce some useful techniques toward a higher dimensional Bieberbach conjecture.

On Beurling's boundary value problem in circle packing

In this survey paper we discuss the problem of characterizing the critical sets of bounded analytic functions in the unit disk of the complex plane. This problem is closely related to the Berger–Nirenberg problem in differential geometry as well as to the problem of describing the zero sets of functions in Bergman spaces. It turns out that for any non-constant bounded analytic function in the unit disk there is always a (essentially) unique “maximal” Blaschke product with the same critical points. These maximal Blaschke products have remarkable properties similar to those of Bergman space inner functions and they provide a natural generalization of the class of finite Blaschke products.