Decompositions of ZMC Graphs and Euler-Ramanujan type identities

Abstract

In this paper, we study finite and infinite decomposition formulas for zero mean curvature (ZMC) graphs in Euclidean, Lorentz--Minkowski, and isotropic (3)-spaces. We first derive new Euler--Ramanujan-type identities that decompose the conjugate of Scherk's first minimal surface into dilated catenoids. Using Weierstrass factorisation and power series methods, we then obtain infinite decompositions for a broad class of isotropic ZMC graphs into helicoids, logarithmoids of revolution, and Enneper surfaces. These results are extended to wider families of ZMC surfaces arising from the López--Ros transformation, Bonnet rotation, and a one-parameter family of metric deformations. We also establish finite decomposition formulas, including analogues of Scherk tower decompositions in Euclidean and isotropic settings, and prove a characterisation theorem for finite decompositions of isotropic minimal surfaces. Finally, we discuss applications to lamellar structures.

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BibTeX RIS

BibTeX

@misc{vasu2026decompositions,
  title = {{Decompositions of ZMC Graphs and Euler-Ramanujan type identities}},
  author = {Priyank Vasu and Sam K Mathew and Rahul Kumar Singh and Rukmini Dey},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2606.11697},
  eprint = {2606.11697},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.11697},
  note = {CC BY-SA 4.0},
}

RIS

TY  - UNPB
TI  - Decompositions of ZMC Graphs and Euler-Ramanujan type identities
AU  - Priyank Vasu
AU  - Sam K Mathew
AU  - Rahul Kumar Singh
AU  - Rukmini Dey
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2606.11697
UR  - https://arxiv.org/abs/2606.11697
AB  - In this paper, we study finite and infinite decomposition formulas for zero mean curvature (ZMC) graphs in Euclidean, Lorentz--Minkowski, and isotropic (3)-spaces. We first derive new Euler--Ramanujan-type identities that decompose the conjugate of Scherk's first minimal surface into dilated catenoids. Using Weierstrass factorisation and power series methods, we then obtain infinite decompositions for a broad class of isotropic ZMC graphs into helicoids, logarithmoids of revolution, and Enneper surfaces. These results are extended to wider families of ZMC surfaces arising from the López--Ros transformation, Bonnet rotation, and a one-parameter family of metric deformations. We also establish finite decomposition formulas, including analogues of Scherk tower decompositions in Euclidean and isotropic settings, and prove a characterisation theorem for finite decompositions of isotropic minimal surfaces. Finally, we discuss applications to lamellar structures.
ER  -

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