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Minimal surfaces over the Pitot quadrilaterals

Vladimir Dragović, David Kalaj

TL;DR

The paper develops an explicit framework to construct Scherk-type minimal graphs over Pitot quadrilaterals (including concave ones) by mapping the unit disk harmonically onto the quadrilateral with a dilatation given by a squared Möbius automorphism, and then using this data in the Enneper–Weierstrass representation to obtain a minimal surface Σ with Scherk-type asymptotics. It introduces a canonical surface Σ^diamond with the same boundary behavior and proves a sharp curvature comparison: at the harmonic center, Σ^diamond uniquely maximizes the absolute Gaussian curvature among all bounded minimal graphs sharing the same normal and mixed derivative. The approach provides a complete, constructive description of Scherk-type minimal graphs over all Pitot quadrilaterals and extends classical Scherk theory beyond convex domains. These results yield explicit formulas, a two-variable reduction, and a curvature-maximization principle that deepen the understanding of minimal graphs over polygonal domains.

Abstract

We develop a fully explicit framework for constructing Scherk-type minimal graphs over the Pitot quadrilaterals (i.e. such that the two pairs of opposite sides have the same total length). For any Pitot quadrilateral \(Q\), we first produce a harmonic diffeomorphism of the unit disk onto \(Q\), whose dilatation is the square of a Möbius automorphism determined directly by the vertices of \(Q\). Using this map as the Weierstrass data, we obtain a minimal graph \(Σ\) whose Gauss map is a univalent Möbius transformation and whose height function exhibits alternating blow-up behavior along opposite sides of \(Q\), mirroring the classical Scherk surfaces. We further construct an associated canonical surface \(Σ^\diamond\), with the same boundary asymptotics, and prove a sharp curvature comparison theorem: at the harmonic center of \(Q\), among all bounded minimal graphs with matching normal direction and mixed derivative, \(Σ^\diamond\) uniquely maximizes the absolute Gaussian curvature. This provides a complete and constructive description of Scherk-type minimal graphs over all, both convex or concave, Pitot quadrilaterals.

Minimal surfaces over the Pitot quadrilaterals

TL;DR

The paper develops an explicit framework to construct Scherk-type minimal graphs over Pitot quadrilaterals (including concave ones) by mapping the unit disk harmonically onto the quadrilateral with a dilatation given by a squared Möbius automorphism, and then using this data in the Enneper–Weierstrass representation to obtain a minimal surface Σ with Scherk-type asymptotics. It introduces a canonical surface Σ^diamond with the same boundary behavior and proves a sharp curvature comparison: at the harmonic center, Σ^diamond uniquely maximizes the absolute Gaussian curvature among all bounded minimal graphs sharing the same normal and mixed derivative. The approach provides a complete, constructive description of Scherk-type minimal graphs over all Pitot quadrilaterals and extends classical Scherk theory beyond convex domains. These results yield explicit formulas, a two-variable reduction, and a curvature-maximization principle that deepen the understanding of minimal graphs over polygonal domains.

Abstract

We develop a fully explicit framework for constructing Scherk-type minimal graphs over the Pitot quadrilaterals (i.e. such that the two pairs of opposite sides have the same total length). For any Pitot quadrilateral , we first produce a harmonic diffeomorphism of the unit disk onto , whose dilatation is the square of a Möbius automorphism determined directly by the vertices of . Using this map as the Weierstrass data, we obtain a minimal graph whose Gauss map is a univalent Möbius transformation and whose height function exhibits alternating blow-up behavior along opposite sides of , mirroring the classical Scherk surfaces. We further construct an associated canonical surface , with the same boundary asymptotics, and prove a sharp curvature comparison theorem: at the harmonic center of , among all bounded minimal graphs with matching normal direction and mixed derivative, uniquely maximizes the absolute Gaussian curvature. This provides a complete and constructive description of Scherk-type minimal graphs over all, both convex or concave, Pitot quadrilaterals.

Paper Structure

This paper contains 7 sections, 12 theorems, 138 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reduction of the problem to two complex variables $z,w$
  4. The geometric meaning of the parameter $t$
  5. Proofs of Theorem \ref{['teo1']} and Theorem \ref{['teo2']}
  6. The principal part notation.
  7. A proof of Theorem \ref{['etreta']}

Key Result

Theorem 1.1

Let $Q$ be a Pitot quadrilateral with consecutive vertices $b_1,b_2,b_3,b_4$. Then there exists a parameter $p=p(b_1,b_2,b_3,b_4)\in(0,\pi)$, such that the step function $F:[0,2\pi)\to \mathbb{C}$, given by has the Poisson extension $f=P[F]$ to the unit disk $\mathbb{D}$, that is a sense-preserving harmonic mapping with the following properties:

Figures (5)

  • Figure 1: The quadrilateral whose opposite vertices are the foci $-1$ and $1$, and whose remaining vertices lie on the same branch of the hyperbola $|\zeta+1|-|\zeta-1|=2\sin m$, i.e. of $x^{2}\csc^{2} m - y^{2}\sec^{2} m = 1.$
  • Figure 2: The concave quadrilateral $Q(-1,z,1,w)$ for $m=0.3$, $t=0.3$, $s=1$, $c_0= 0.299+ 0.552\imath$.
  • Figure 3: The Scherk type surface over the concave $Q$ with its unit normal $N$ at the center.
  • Figure 4: The convex quadrilateral $Q(-1,z,1,w)$ for $m=0.3$, $t=-0.3$, $s=1$, $c_0= 0.234 +0.255 \imath$.
  • Figure 5: The Scherk type surface over the convex $Q$ with its unit normal $N$ at the center.

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 13 more