Closed surfaces with bounds on their Willmore energy

2010/09/27 by Kuwert, Ernst, Schätzle, Reiner · 1 citation
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1009.5286

Abstract

The Willmore energy of a closed surface in Rⁿ is the integral of its squared mean curvature, and is invariant uner Möbius transformations of Rⁿ. We show that any torus in R³ with energy at most 8 π-delta has a representative under the Möbius action, for which the induced metric and a conformal metric of constant (zero) curvature are uniformly equivalent, with constants depending only on delta>0. An analogous estimate is also obtained for surfaces of fixed genus p ≥ 1 in R³ or R⁴, assuming suitable energy bounds which are sharp for n=3. Moreover the conformal type is controlled in terms of the energy bounds.

Cited by

• Elastic bilayer membranes under confinement -- Existence, regularity, and rigidity

Related

• Minimizers of the Willmore functional under fixed conformal class
• Energy Quantization for Willmore Surfaces and Applications
• Quantization of the Willmore Energy in Riemannian Manifolds
• Rolling spheres and the Willmore energy
• Conformal Willmore Tori in ℝ⁴