Finite Decomposition of Minimal surfaces, Maximal surfaces, Timelike Minimal surfaces and Born-Infeld solitons

TL;DR

This paper demonstrates that various classes of surfaces, including minimal, maximal, timelike minimal surfaces, and Born-Infeld solitons, can be decomposed into finite sums of scaled and translated versions of themselves, revealing new structural properties.

Contribution

It introduces a general method for decomposing height functions of these surfaces into finite sums of similar functions, extending previous results and providing new examples and decompositions.

Findings

Height functions decompose into finite sums of scaled and translated versions.

Decomposition applies to minimal, maximal, timelike minimal surfaces, and Born-Infeld solitons.

New examples of decompositions and foliations are provided.

Abstract

We show that the height function of Scherk's second surface decomposes into a finite sum of scaled and translated versions of itself, using an Euler Ramanujan identity. A similar result appears in R. Kamien's work on liquid crystals where he shows (using an Euler-Ramanujan identity) that the Scherk's first surface decomposes into a finite sum of scaled and translated versions of itself. We give another finite decomposition of the height function of the Scherk's first surface in terms of translated helicoids and scaled and translated Scherk's first surface. We give some more examples, for instance a (complex) maximal surface and a (complex) BI soliton. We then show, using the Weierstrass-Enneper representation of minimal (maximal) surfaces, that one can decompose the height function of a minimal (maximal) surface into finite sums of height functions of surfaces which, upon change of coordinates, turn out to be minimal (maximal) surfaces, each minimal (maximal) w.r.t. to its own new coordinates. We then exhibit a general property of minimal surfaces, maximal surfaces, timelike minimal surfaces and Born-Infeld soliton surfaces that their local height functions $z=Z(x,y)$ split into finite sum of scaled and translated versions of functions of the same form. Upto scaling these new functions are height functions of the minimal surfaces, maximal surfaces, timelike minimal surfaces and Born-Infeld soliton surfaces respectively. Lastly, we exhibit a foliation of ${\mathbb R}^3$ minus certain lines by shifted helicoids (which appear in one of the Euler-Ramanujan identities).