arXiv subjects

Priyank Vasu

Publications and source records attributed to Priyank Vasu.

7 recordsLinked to original sources

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

Superposition of Harmonic Surfaces: Helical Motifs in Lamellar Structures

We study harmonic surfaces in $\mathbb{R}^3$ through the framework of harmonic Enneper immersions and prove a superposition principle for such surfaces. We prove that minimal and maximal surfaces admit a decomposition into harmonic components. Applications include the construction of finite and infinite configurations of helical motifs, an asymptotic analysis via multipole expansions, and the modelling of twist grain boundary phases in lamellar systems.

math.DG↗

arXiv2026

Duality of zero mean curvature surfaces in the Lorentzian Heisenberg group

We study a transformation surface associated with a zero mean curvature surface in the three-dimensional Heisenberg group with respect to two left-invariant semi-Riemannian metrics. We investigate the duality and prove that the transformation surface also has zero mean curvature. Furthermore, we derive the Sym formula for the dual surface in both metric cases.

math.DG↗

arXiv2026

Harmonic Enneper Immersion in $\mathbb{R}^3$

We present a method for constructing harmonic immersions in $\mathbb{R}^3$, known as the Enneper-type representation. We also prove that any harmonic immersion in $\mathbb{R}^3$ can be obtained using this approach. Furthermore, we determine the number of non-planar rotational harmonic immersions in $\mathbb{R}^3$ that connect two coaxial circles in parallel planes, where both circles have the same radius $r > 0$ and are separated by a distance $l > 0$.

math.DG↗

arXiv2025

Decompositions of Scherk-Type Zero Mean Curvature Surfaces

In this paper, by using a special Euler-Ramanujan identity and the idea of Wick rotation, we show that a one-parameter family of solutions to the zero mean curvature equation in Lorentz-Minkowski $3$-space $\mathbb E_1^3$, namely Scherk-type zero mean curvature surfaces, can be expressed as an infinite superposition of dilated helicoids. Further, we also obtain different finite decompositions for these surfaces. We end this paper with an application of these decompositions to formulate maximal codimension 2 surfaces into finite and infinite "sums" of weakly untrapped and *-surfaces in Lorentz-Minkowski 4-space.

math.DG↗

arXiv2025

Modular Surfaces in Lorentz-Minkowski 3-Space: Curvature and Applications

In this paper, we study the relation of the sign of the Gaussian and mean curvature of modular surfaces in Lorentz-Minkowski $3$-space to the zeroes of the associated complex analytic functions and its derivatives. Further, we completely classify zero Gaussian curvature modular surfaces. Next we show non-existence of non-planar maximal modular surfaces, characterize CMC modular surfaces, analyze asymptotic behaviour of Gaussian curvature of complete modular graphs and the Hessian of their height functions and lastly as application, demonstrate how modular surfaces can be realised as integral surfaces of some conformal field theories and non-linear sigma models.

math.DG↗

arXiv2025

Timelike minimal surface in $\mathbb{E}^3_1$ with arbitrary ends

In this paper, we show the existence of a timelike minimal surface with an arbitrary number of weak complete ends. Then, we discuss the asymptotic behaviour of the simple ends and the topology of the singularity set of the constructed timelike minimal surface.

math.DG↗