Extendability of conformal structures on punctured surfaces

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 $\|\nabla f\|_{L^\infty}$ is finite and the second fundamental form of $f$ is in $L^2$, we show that there exists a homeomorphism $\phi:D\to D$ such that $f\circ\phi$ is a branched $W^{2,2}$-conformal immersion from the Euclidean unit disk $D$ into $\mathbb{R}^n$.

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