Notes on Minimal Surfaces and Differential Forms II -- Bateman's Wave Functions

A remarkable result of differential topology that should have widespread engineering application (but an idea that has received almost no attention in practice) is that there exists a well-defined set of time dependent vector fields that generate minimal surfaces in 4 dimensions. Sophus Lie proved that any holomorphic function could be considered as a complex curve, or a position vector in 4 dimensions. These vectors always generate conjugate minimal surfaces. The utility of this class of vector fields was first brought to this author's attention by the small 1914 monograph of H. Bateman entitled Electrical and Optical Wave Motion, Dover (1955). In this monograph Bateman published (without reference to the theory of minimal surfaces) some extraordinary results of such complex vector fields, including examples of solutions to Maxwell's equations that emulate propagating singular strings (not plane waves).


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