Rigidity and non-rigidity results for conformal immersions

TL;DR

This paper investigates rigidity properties of conformal immersions, proving a quantitative rigidity result for certain minimizers, and providing explicit counterexamples in other cases, highlighting the nuanced behavior of Willmore surfaces.

Contribution

It establishes a new rigidity result for Willmore minimizers among projective planes and constructs counterexamples demonstrating limits of rigidity in other settings.

Findings

Rigidity holds for minimizers of the Willmore functional among projective planes.

Counterexamples show rigidity fails in codimension one and for large Willmore energies.

Explicit constructions illustrate the splitting-off of Enneper surfaces during blow-up processes.

Abstract

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].