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 Ω 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.