arXiv · 1604.04071
This article presents a clear proof of the Riemann Mapping Theorem via Riemann's method, uncompromised by any appeals to topological intuition.
Robert E. Greene, Kang-Tae Kim. 2016-04-14. The Riemann mapping theorem from Riemann's viewpoint. https://arxiv.org/abs/1604.04071
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We study the two-dimensional Matkowski--Sut\^o equation, which asks for two quasi-arithmetic means whose sum is twice the arithmetic mean, in two settings. For holomorphic injective generators with convex images on a convex domain in the complex plane, the solutions are exactly the affine pairs and the exponential pairs with a nonzero complex exponent, up to affine changes of the generators. The admissible exponents depend on the shape of the domain and are described by a curvature criterion for its boundary. In the monotone-operator framework of T\'oth, we construct an infinite-dimensional family of non-affine shear pairs on the whole plane. Their generators are strictly increasing in the sense of monotone operators and need not be differentiable. These pairs solve the weighted equation for any number of variables. The rigidity of the one-dimensional problem, due to Dar\'oczy and P\'ales, persists under holomorphy but not under monotonicity.
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