Minimal surfaces over the Pitot quadrilaterals

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.