QUANTITATIVE RIGIDITY RESULTS FOR CONFORMAL IMMERSIONS

In this paper we prove several quantitative rigidity results for conformal immersions of surfaces in 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,W-2-norm. Moreover, we apply these results to prove a corresponding rigidity result for complete, connected and non-compact surfaces.