Second order Contact of Minimal Surfaces

Abstract

The minimal surface equation $Q$ in the second order contact bundle of $R^3$, modulo translations, is provided with a complex structure and a canonical vector-valued holomorphic differential form $Omega$ on $Q\0$. The minimal surfaces $M$ in $R^3$ correspond to the complex analytic curves $C$ in $Q$, where the derivative of the Gauss map sends $M$ to $C$, and $M$ is equal to the real part of the integral of $Ω$ over $C$. The complete minimal surfaces of finite topological type and with flat points at infinity correspond to the algebraic curves in $Q$.