Extendability of conformal structures on punctured surfaces

Abstract

For a smooth immersion $f$ from the punctured disk $D\backslash \{0\}$ into ${\mathbb{R}}^n$ extendable continuously at the puncture, if its mean curvature is square integrable and the measure of $f(D)\cap B_{r_k}=o(r_k)$ for a sequence $r_k\to 0$, we show that the Riemannian surface $(D_r\backslash \{0\},g)$ where $g$ is the induced metric is conformally equivalent to the unit Euclidean punctured disk, for any $r\in (0,1)$. For a locally $W^{2,2}$ Lipschitz immersion $f$ from the punctured disk $D_2\backslash \{0\}$ into ${\mathbb{R}}^n$, if $\Vert \nabla f{\Vert }_{L^{\infty }}$ is finite and the second fundamental form of $f$ is in $L^2$, we show that there exists a homeomorphism $\varphi :D\to D$ such that $f\circ \varphi$ is a branched $W^{2,2}$-conformal immersion from the Euclidean unit disk $D$ into ${\mathbb{R}}^n$.

19 pages

Cited by in corpus (1)

• Gromov-Hausdorff convergence theory of surfaces