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Quantitative rigidity results for conformal immersions
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 136, Number 5, October 2014
- pp. 1409-1440
- 10.1353/ajm.2014.0033
- Article
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In this paper we prove several quantitative rigidity results for conformal immersions of surfaces in ${\Bbb R}^n$
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.