Quantitative rigidity results for conformal immersions (Q2927719)

The authors prove a number of stability results for conformal immersions of surfaces in \(\mathbb{R}^n\). These results are of the following type: If the conformal immersion \(f\) is close to a certain prototype surface \(g\) with respect to total curvature, then it is also close to that surface in a suitable Möbius invariant norm. The prototype surfaces (round sphere, Clifford torus, inverted catenoid, inverted Enneper minimal surface, and inverted Chen minimal graph) differ by their dimension, genus, and branches of the immersion. The proofs are largely based on convergence results for sequences of conformal immersions which are interesting in their own right.NEWLINENEWLINESome proofs are only valid under the assumption of the Willmore conjecture which apparently was still open at the time of writing. The recent proof of Willmore's conjecture by \textit{F. C. Marques } and \textit{A. Neves } [Ann. Math. (2) 179, No. 2, 683--782 (2014; Zbl 1297.49079)] settles these cases.