On conformal surfaces of annulus type

TL;DR

This paper investigates the properties of conformal maps from annular regions in the plane into Euclidean space, focusing on the associated moving frame and its Dirichlet energy, with applications to geometric analysis.

Contribution

It introduces a new analysis of the moving frame associated with conformal annular maps and explores its Dirichlet energy, providing novel insights and applications in geometric analysis.

Findings

The moving frame satisfies a specific differential equation involving the Hodge star operator.

The Dirichlet energy of the frame can be characterized and utilized in applications.

The study offers new tools for understanding conformal surfaces of annulus type.

Abstract

Let $a>b>0$ and $f$ be a conformal map from $B_a\setminus B_b\subseteq R^2$ into $\R^n$, with $|\nabla f|^2=2e^{2u}$. Then $(e_1, e_2)$ with $e_1=e^{-u}\frac{\partial f}{\partial r},$ and $e_2=r^{-1}e^{-u}\frac{\partial f}{\partial\theta}$ is a moving frame on $f(B_a\setminus B_b)$. It satisfies the following equation $$d\star<de_1, e_2>=0,$$ where $\star$ is the Hodge star operator on $R^2$ with respect to the standard metric. We will study the Dirichret energy of this frame and give some applications.