Explicit Bj\"orling Surfaces with Prescribed Geometry

TL;DR

This paper introduces a novel method for explicitly constructing regular minimal surfaces in Euclidean space with controlled geometry, based on the Björling formula applied to a broad class of planar curves.

Contribution

It provides a new explicit construction technique for minimal surfaces using the Björling formula, enabling controlled geometric properties for entire complex plane domains.

Findings

Explicit formulas for minimal surfaces derived from planar curves.

A wide class of curves can generate minimal surfaces with prescribed geometry.

Numerous examples demonstrating the method's effectiveness.

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 $(x(t), y(t))$ one 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 Bj\"orling formula applied to $c(t)$ and $n(t)$ can be explicitly evaluated. We give many examples.