Quantitative rigidity results for conformal immersions

Abstract

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.

27 pages, to appear in Amer. J. Math

References in corpus (2)

• Bubble tree of a class of conformal mappings and applications to Willmore functional
• Optimal rigidity estimates for nearly umbilical surfaces in arbitrary codimension