A problem for isolated singularities of surfaces

Let $E\subset\mathbf R^n$ be a set of dimension $2$ with an isolated singular point $0$, such that $E-\{0\}$ is an orientable $\mathcal C^2$ submanifold of $\mathbf R^n$ of dimension $2$, and hence a Riemann surface with the conformal structure induced by the ambient metric. We study the extension o