Complete bounded holomorphic curves immersed in ℂ² with arbitrary genus

Martin, Francisco; Umehara, Masaaki; Yamada, Kotaro

doi:10.1090/S0002-9939-09-09953-5

Complete bounded holomorphic curves immersed in $\mathbb {C}^2$ with arbitrary genus
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by Francisco Martin, Masaaki Umehara and Kotaro Yamada PDF

Proc. Amer. Math. Soc. 137 (2009), 3437-3450 Request permission

Abstract:

Recently, a complete holomorphic immersion of the unit disk $\mathbb {D}$ into $\mathbb {C}^2$ whose image is bounded was constructed by the authors. In this paper, we shall prove the existence of complete holomorphic null immersions of Riemann surfaces with arbitrary genus and finite topology whose image is bounded in $\mathbb {C}^2$.

As an analogue to the above construction, we also give a new method to construct complete bounded minimal immersions (resp. weakly complete maximal surfaces) with arbitrary genus and finite topology in Euclidean 3-space (resp. Lorentz-Minkowski 3-spacetime).

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