# Journal of the Ramanujan Mathematical Society

Volume 38, Issue 3, September 2023 pp. 225–236.

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

**Authors**:
Rukmini Dey, Kohinoor Ghosh and Sidharth Soundararajan

**Author institution:**
I.C.T.S.-T.I.F.R., Bangalore.

**Summary: **
This paper deals with decomposition of height functions of various zero-mean
curvature surfaces into a finite sum of scaled and translated versions of
themselves. There are various ways of accomplishing this. 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. 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 ℝ³ minus
certain lines by shifted helicoids (which appear in one of the
Euler-Ramanujan identities).

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