Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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
$\mathbb{R}^n$.

12
The categories of this paper are: nucl-th
31
The categories of this paper are: ao-sci
13
The categories of this paper are: quant-ph
17
The categories of this paper are: q-fin
21
The categories of this paper are: adap-org
5
The categories of this paper are: gr-qc
1
The categories of this paper are: cs
36
The categories of this paper are: cmp-lg
9
The categories of this paper are: nlin
22
The categories of this paper are: chao-dyn
20
The categories of this paper are: acc-phys
34
The categories of this paper are: chem-ph
8
The categories of this paper are: hep-ex
26
The categories of this paper are: bayes-an
15
The categories of this paper are: nucl-ex
4
The categories of this paper are: cond-mat