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 .

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.