Minimal surfaces over the Pitot quadrilaterals

TL;DR

This paper presents an explicit method for constructing Scherk-type minimal graphs over Pitot quadrilaterals, including a harmonic map approach, curvature comparison, and a complete classification for all such quadrilaterals.

Contribution

It introduces a fully explicit framework for minimal graphs over Pitot quadrilaterals using harmonic diffeomorphisms and provides a sharp curvature comparison theorem for these surfaces.

Findings

Constructed harmonic diffeomorphisms with explicit dilatation

Developed minimal graphs with alternating blow-up behavior

Proved maximal Gaussian curvature at harmonic center

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 M\"obius automorphism determined directly by the vertices of \(Q\). Using this map as the Weierstrass data, we obtain a minimal graph \(\Sigma\) whose Gauss map is a univalent M\"obius 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 \(\Sigma^\diamond\), 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, \(\Sigma^\diamond\) 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.