Rigidity and non-rigidity results for conformal immersions
Copyright policy )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 $\mathbb{R}^n$ with $n\ge 4$. 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 [LaNg13].
Minor modifications, to appear in Adv. Math
- Karlsruhe Institute of Technology Germany
- EBERHARD KARLS UNIVERSITAET TUEBINGEN Germany
- University of Tübingen Germany
ddc:510, Mathematics - Differential Geometry, Geometric measure and integration theory, integral and normal currents in optimization, rigidity results, Methods of global Riemannian geometry, including PDE methods; curvature restrictions, 510, Surfaces in Euclidean and related spaces, Mathematics - Analysis of PDEs, Differential Geometry (math.DG), FOS: Mathematics, conformal parametrization, Conformal differential geometry, Willmore functional, Mathematics, info:eu-repo/classification/ddc/510, Analysis of PDEs (math.AP)
ddc:510, Mathematics - Differential Geometry, Geometric measure and integration theory, integral and normal currents in optimization, rigidity results, Methods of global Riemannian geometry, including PDE methods; curvature restrictions, 510, Surfaces in Euclidean and related spaces, Mathematics - Analysis of PDEs, Differential Geometry (math.DG), FOS: Mathematics, conformal parametrization, Conformal differential geometry, Willmore functional, Mathematics, info:eu-repo/classification/ddc/510, Analysis of PDEs (math.AP)
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