A new approach to the Fraser-Li conjecture with the Weierstrass representation formula

TL;DR

This paper introduces a new PDE approach to the Fraser-Li conjecture by linking boundary conditions of free boundary minimal surfaces to Weierstrass data and analyzing the Gauss map of minimal annuli.

Contribution

It provides a sufficient condition for boundary curves via Weierstrass data, proves the injectivity of the Gauss map for embedded minimal annuli, and reformulates the Fraser-Li conjecture as a Gauss map problem.

Findings

Boundary condition expressed without integration using Weierstrass data

Gauss map of embedded free boundary minimal annulus is injective

New PDE perspective on the Fraser-Li conjecture

Abstract

In this paper, we provide a sufficient condition for a curve on a surface in $\mathbb{R}^3$ to be given by an orthogonal intersection with a sphere. This result makes it possible to express the boundary condition entirely in terms of the Weierstrass data without integration when dealing with free boundary minimal surfaces in a ball $\mathbb{B}^3$. Moreover, we show that the Gauss map of an embedded free boundary minimal annulus is one to one. By using this, the Fraser-Li conjecture can be translated into the problem of determining the Gauss map. On the other hand, we show that the Liouville type boundary value problem in an annulus gives some new insight into the structure of immersed minimal annuli orthogonal to spheres. It also suggests a new PDE theoretic approach to the Fraser-Li conjecture.