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Surfaces with quadratic support function of harmonic type

Armando M. V. Corro, Carlos M. C. Riveros, José L. Teruel

TL;DR

This work introduces surfaces in $\\mathbb{R}^3$ called HQSF-surfaces that satisfy the relation $2\\Psi H+(c\\Psi e^{2\\mu}+\\Lambda-\\Psi^2)K=0$, generalizing QSF-surfaces. It develops a Weierstrass-type representation for HQSF-surfaces depending on three holomorphic data $(g,f_1,f_2)$ and provides explicit expressions for the geometric invariants. It also provides a full rotation classification, giving an explicit parametric form of rotational HQSF-surfaces and analyzing singularities and completeness across parameter regimes. The framework yields a constructive holomorphic approach to generating HQSF-surfaces and clarifies how the harmonic term $\\mu$ influences curvature and singular behavior. These results extend Laguerre-type and DSGWH surface families and enable new explicit examples and potential applications in surface theory.

Abstract

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.

Surfaces with quadratic support function of harmonic type

TL;DR

This work introduces surfaces in called HQSF-surfaces that satisfy the relation , generalizing QSF-surfaces. It develops a Weierstrass-type representation for HQSF-surfaces depending on three holomorphic data and provides explicit expressions for the geometric invariants. It also provides a full rotation classification, giving an explicit parametric form of rotational HQSF-surfaces and analyzing singularities and completeness across parameter regimes. The framework yields a constructive holomorphic approach to generating HQSF-surfaces and clarifies how the harmonic term influences curvature and singular behavior. These results extend Laguerre-type and DSGWH surface families and enable new explicit examples and potential applications in surface theory.

Abstract

In this paper, we study oriented surfaces S in , 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.
Paper Structure (3 sections, 4 theorems, 82 equations, 23 figures)

This paper contains 3 sections, 4 theorems, 82 equations, 23 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Surfaces with quadratic support function of harmonic type

Key Result

Lemma 1

If $f_1, f_2, g_1, g_2:\mathbb{C} \rightarrow \mathbb{C}$ are holomorphic functions of $z=u_1+iu_2$, such that $\,\langle f_1,g_1\rangle+\langle f_2,g_2\rangle=0,$ then where $c_i$ are real constants and $z_1 \in \mathbb{C}.$

Figures (23)

  • Figure 1: HQSF-surface
  • Figure 2: HQSF-surface
  • Figure 3: HQSF-surface
  • Figure 4: HQSF-surface
  • Figure 5: HQSF-surface
  • ...and 18 more figures

Theorems & Definitions (21)

  • Lemma 1
  • Theorem 1
  • Definition 1
  • Theorem 2
  • proof
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • ...and 11 more