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Search: MSC category 30C55 ( General theory of univalent and multivalent functions )

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1. CJM Online first

Roth, Oliver
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 Pontryagin's 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 mappings for a wide class of extremal problems over the set of normalized biholomorphic mappings on the unit ball in $\mathbb{C}^n$.

Keywords:univalent function, Loewner's equation
Categories:32H02, 30C55, 49K15

2. CJM 1999 (vol 51 pp. 470)

Bshouty, D.; Hengartner, W.
Exterior Univalent Harmonic Mappings With Finite Blaschke Dilatations
In this article we characterize the univalent harmonic mappings from the exterior of the unit disk, $\Delta$, onto a simply connected domain $\Omega$ containing infinity and which are solutions of the system of elliptic partial differential equations $\fzbb = a(z)f_z(z)$ where the second dilatation function $a(z)$ is a finite Blaschke product. At the end of this article, we apply our results to nonparametric minimal surfaces having the property that the image of its Gauss map is the upper half-sphere covered once or twice.

Keywords:harmonic mappings, minimal surfaces
Categories:30C55, 30C62, 49Q05

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