Surfaces with quadratic support function of harmonic type

In this paper, we study oriented surfaces S in $\mathbb{R}^3$, called Surfaces with quadratic support function of harmonic type (in short HQSF-surfaces), these surfaces generalize the QSF-surfaces. We obtain a Weierstrass type representation for the HQSF-surfaces which depends on three holomorphic functions. Moreover, we classify the HQSF-surfaces of rotation.

💡 Research Summary

The paper introduces a new class of oriented surfaces in Euclidean three‑space, called HQSF‑surfaces (Surfaces with Quadratic Support Function of Harmonic type). An HQSF‑surface is defined by the relation

  2 Ψ H + (c Ψ e^{2µ} + Λ − Ψ²) K = 0,

where H and K are the mean and Gaussian curvatures, Ψ(p)=⟨p,N(p)⟩ is the support function, Λ(p)=⟨p,p⟩ is the quadratic distance function, µ is a harmonic function with respect to the third fundamental form, and c∈ℝ is a constant. This equation generalizes several previously studied families, such as QSF‑surfaces (c Ψ e^{2µ}=0) and various Weingarten and Laguerre minimal surfaces.

The first major result (Theorem 2) is a Weierstrass‑type representation: every HQSF‑surface can be locally parametrized by three holomorphic functions g, f₁, f₂ defined on a Riemann surface. Setting

 A = |f₁|² + c |f₂|², T = 1 + |g|², h = A/T,

the support and distance functions become Ψ = 2h T and Λ = |∇h|²|g′|² − 4Rh T, where R is an explicit combination of f₁, f₂, g and their derivatives. Substituting these expressions into the curvature relation yields the compatibility condition

 µ = ln(2 |f₁ f₂′ − f₂ f₁′| · |g′|).

Consequently the immersion X(z) is given by the explicit formula (3.2) in the paper, which depends on the three holomorphic data. This representation extends the classical two‑function Weierstrass formula for QSF‑surfaces and shows that the harmonic term e^{2µ} can be encoded holomorphically.

The second major contribution is a complete classification of HQSF‑surfaces possessing rotational symmetry. Assuming the Gauss map g(w)=w and a radial function h(w)=r(|w|), the authors reduce the problem to an ordinary differential equation (3.26) for a complex function f₁. The characteristic equation λ² − (c c₁ + c₂)λ + (c₁c₂ − |z₁|²)=0 has discriminant Ω = (c c₁ − c₂)² + 4c|z₁|². Three cases arise:

1. Ω > 0 (two distinct real roots): f₁ and f₂ are linear combinations of exponentials e^{λz/(2c)}; the resulting profile curves A(u₁), B(u₁) are given by formulas (3.18)–(3.20).

2. Ω < 0 (complex conjugate roots): f₁ and f₂ involve sine and cosine of √{−Ω} z/(2c); the profile curves are expressed in terms of trigonometric functions (3.30)–(3.31).

3. Ω = 0 (double root): f₁ is linear in z, leading to polynomial expressions for the profile (3.32)–(3.33).

In each case the surface can be written as

 X_i(u₁,u₂) = (A_i(u₁) cos u₂, A_i(u₁) sin u₂, B_i(u₁)),

with A_i and B_i computed from the corresponding holomorphic data. The authors provide explicit formulas for A_i, B_i (equations (3.15)–(3.17)) and discuss the regularity conditions.

A series of examples illustrates the theory. Examples 1–5 select simple holomorphic functions (e.g., f₁(z)=z, f₂(z)=e^{z}, g(z)=z²) and various constants c to produce visually distinct HQSF‑surfaces. Examples 6–11 focus on rotational cases, showing how different choices of the parameters (a₁, a₂, c₁, c₂, z₁) generate profile curves that intersect the axis of rotation multiple times, leading to isolated singularities or circles of singular points. The figures demonstrate that HQSF‑surfaces can exhibit far richer singular behavior than classical minimal or Laguerre‑minimal surfaces.

Overall, the paper makes three substantive contributions: (i) it defines a new, broader class of Weingarten‑type surfaces incorporating a harmonic term; (ii) it derives a three‑function holomorphic representation that generalizes earlier Weierstrass formulas; and (iii) it provides a full classification of rotational HQSF‑surfaces, together with a detailed analysis of their singular sets. The work bridges complex analysis and differential geometry, offering tools that may be applied to further studies of Laguerre geometry, affine differential geometry, and integrable surface theory.