Unicity Results for Gauss Maps of Minimal Surfaces Immersed in R^m



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The purpose of this dissertation is to discuss the theory of holomorphic curves in order to study value distributions of (generalized) Gauss maps of complete minimal surfaces immersed in R^m. The study was initiated by S.S. Chern and R. Osserman in the 1960s. Since then, it has been developed by F. Xavier, H. Fujimoto, M. Ru, etc. In this dissertation, we prove a unicity theorem for two conformally diffeomorphic complete minimal surfaces immersed in R^m whose generalized Gauss maps f and g agree on the pre-image ⋃_{j=1}^q f^{-1}(H_j) for given hyperplanes H_j (1≤ j ≤ q) in P^{m-1}(C) located in general position, under the assumption that ⋂_{j=1}^{k+1} f^{-1}(H_{i_j}) = ∅. In the case when k=m-1, the result obtained gives an improvement of the earlier result of Fujimoto [13].



Unicity Theorem, Gauss maps, Minimal surfaces, Hyperplanes, Complex geometry