On 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.