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Minimal and Maximal Surfaces Generated by the Holomorphic Function log(z)

The holomorphic function in the Gauss plane generates a minimal surface in Euclidean space and a minimal time-like surface in Minkowski space ; the same holomorphic function in the Lorentz plane generates a maximal space-like surface in .
The immersion of the Minkowski space in is used to present the minimal time-like surface and the maximal space-like surface in .

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Any minimal surface in Euclidean space or any minimal time-like surface in Minkowski space is generated by a holomorphic function in the Gauss plane ; any maximal space-like surface in is generated by a holomorphic function in the Lorentz plane .
G. Ganchev, "Canonical Weierstrass Representation of Minimal Surfaces in Euclidean Space," arxiv.org/abs/0802.2374.
G. Ganchev, "Canonical Weierstrass Representation of Minimal and Maximal Surfaces in the Three-Dimensional Minkowski Space," arxiv.org/abs/0802.2632.
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