Sam K Mathew | Searcharxiv

arXiv2026

Decompositions of ZMC Graphs and Euler-Ramanujan type identities

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 L\'opez--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.

math.DG↗

arXiv2026

Minimal Graph Transformations and their Classification

This paper presents a complete classification of minimal graph surfaces that admit graphical transformations into other minimal surfaces. These transformations are functions that map the height function of a minimal graph surface to another height function, which also describes a minimal graph surface. While trivial maps such as translations and reflections exist, we formulate and solve the Non-Trivial Minimal Graph Transformation Problem, governed by a coupled system of partial differential equations. A central result establishes the rigorous equivalence of this original system to a modified problem for a harmonic function. Through a complex variable approach and a weakening technique, the analysis is reduced to solving a fundamental ordinary differential equation parameterized by a real constant k. The explicit integration of this ordinary differential equation involves various elliptic integrals and identities of elliptic functions. Solving the ordinary differential equation for the three cases: when the constant k equals zero, when k is greater than zero, and when k is less than zero yields the full classification of all admissible surfaces and their associated transformations. This process yields several classes of minimal surfaces that, to the best of the author's knowledge, constitute new families of minimal surfaces.

math.DG↗