Explicit Björling Surfaces with Prescribed Geometry

Abstract

We develop a new method to construct explicit regular minimal surfaces in Euclidean space that are defined on the entire complex plane with controlled geometry. More precisely, we show that for a large class of planar curves (x(t),y(t)), we can find a third coordinate z(t) and normal fields n(t) along the space curve c(t)=(x(t),y(t),z(t)) so that the Björling formula applied to c(t) and n(t) can be explicitly evaluated. We give many examples.