Deformation of minimal surfaces with planar curvature lines

TL;DR

This paper provides a new analytical approach to minimal surfaces with planar curvature lines, deriving their metric functions, parametrizations, and demonstrating a continuous deformation among all such surfaces.

Contribution

It introduces an analytical method to compute metric functions and parametrizations, revealing a continuous deformation among minimal surfaces with planar curvature lines.

Findings

Existence of a single continuous deformation between all such surfaces

Derivation of metric functions and Weierstrass data

Identification of axial directions for these surfaces

Abstract

Minimal surfaces with planar curvature lines are classical geometric objects, having been studied since the late 19th century. In this paper, we revisit the subject from a different point of view. After calculating their metric functions using an analytical method, we recover the Weierstrass data, and give clean parametrizations for these surfaces. Then, using these parametrizations, we show that there exists a single continuous deformation between all minimal surfaces with planar curvature lines. In the process, we establish the existence of axial directions for these surfaces.