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arXiv · 2606.11697

Decompositions of ZMC Graphs and Euler-Ramanujan type identities

Priyank Vasu·Sam K Mathew·Rahul Kumar Singh·Rukmini Dey
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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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Priyank Vasu, Sam K Mathew, Rahul Kumar Singh, Rukmini Dey. 2026-08-31. Decompositions of ZMC Graphs and Euler-Ramanujan type identities. https://arxiv.org/abs/2606.11697

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