Minimal and Maximal Surfaces Generated by the Holomorphic Function log(z)
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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 
.
Contributed by: Georgi Ganchev and Radostina Encheva (March 2011)
Open content licensed under CC BY-NC-SA
Snapshots
Details
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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