An approach for minimal surface family passing a curve

TL;DR

This paper develops a method to identify minimal surfaces passing through a specific curve in three-dimensional space, utilizing the Frenet frame and isothermal parameters, with explicit examples involving circles and helices.

Contribution

It introduces a new approach to characterize minimal surfaces passing a given curve using Frenet frames and provides explicit parametric forms for specific cases.

Findings

Derived necessary and sufficient conditions for minimal surfaces passing a curve

Obtained parametric representations for surfaces passing a circle and a helix

Enhanced understanding of minimal surface families in 3D geometry

Abstract

We investigate minimal surfaces passing a given curve in $R^{3}$. Using the Frenet frame of a given curve and isothermal parameter, we derive the necessary and sufficient condition for minimal surface. Also we derive the parametric representation of two minimal surface families passing a circle and a helix as examples.