Extendability of conformal structures on punctured surfaces

TL;DR

This paper investigates conditions under which conformal structures on punctured surfaces can be extended or regularized, showing conformal equivalence to the punctured disk and existence of conformal parametrizations under certain integrability and regularity conditions.

Contribution

It establishes new criteria for extending conformal structures on punctured surfaces and constructing conformal parametrizations for Lipschitz immersions with finite second fundamental form.

Findings

Conformal equivalence to the punctured disk under mean curvature and measure conditions.

Existence of a homeomorphism making the immersion conformal with controlled regularity.

Extension results for conformal structures on surfaces with specific curvature and measure properties.

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 $\|\nabla f\|_{L^\infty}$ 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 $\mathbb{R}^n$.