  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
$\mathbb{R}^n$.
