  In this paper we prove several quantitative rigidity results for conformal
immersions of surfaces in $\mathbb{R}^n$ with bounded total curvature. We show
that (branched) conformal immersions which are close in energy to either a
round sphere, a conformal Clifford torus, an inverted catenoid, an inverted
Enneper's minimal surface or an inverted Chen's minimal graph must be close to
these surfaces in the $W^{2,2}$-norm. Moreover, we apply these results to prove
a corresponding rigidity result for complete, connected and non-compact
surfaces.
