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 $\left(x\right(t),y(t\left)\right)$, we can find a third coordinate $z\left(t\right)$ and normal fields $n\left(t\right)$ along the space curve $c\left(t\right)=\left(x\right(t),y(t),z(t\left)\right)$ so that the Björling formula applied to $c\left(t\right)$ and $n\left(t\right)$ can be explicitly evaluated. We give many examples.