Rigidity and non-rigidity results for conformal immersions

In this paper we show a quantitative rigidity result for the minimizer of the Willmore functional among all projective planes in RnRn with n≥4n≥4. We also construct an explicit counterexample to a corresponding rigidity result in codimension one, by showing that an Enneper surface might split-off during a blow-up process. For conformal immersions of spheres with large enough Willmore energies, we construct explicit counterexamples to a quantitative rigidity result and this complements the recently obtained rigidity results in [14].