Function-theoretic Properties for the Gauss Maps of Various Classes of Surfaces

Kawakami, Yu

Canadian journal of mathematics, Tome 67 (2015) no. 6, pp. 1411-1434

Résumé

We elucidate the geometric background of function-theoretic properties for the Gauss maps of several classes of immersed surfaces in three-dimensional space forms, for example, minimal surfaces in Euclidean three-space, improper affine spheres in the affine three-space, and constant mean curvature one surfaces and flat surfaces in hyperbolic three-space. To achieve this purpose, we prove an optimal curvature bound for a specified conformal metric on an open Riemann surface and give some applications. We also provide unicity theorems for the Gauss maps of these classes of surfaces.

Détail

DOI : 10.4153/CJM-2015-008-5

Mots-clés : 53C42, 30D35, 30F45, 53A10, 53A15, Gauss map, minimal surface, constant mean curvature surface, front, ramification, omitted value, the Ahlfors island theorem, unicity theorem

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