Energy quantization of Willmore surfaces at the boundary of the moduli space

Abstract

We establish an energy quantization result for sequences of Willmore surfaces when the underlying sequence of Riemann surfaces is degenerating in the moduli space. We notably exhibit a new residue which quantifies the potential loss of energy in collar regions. Thanks to these residues, we also establish the compactness (modulo the action of the Möbius group of conformal transformations of R 3 ∪ {∞}) of the space of Willmore immersions of any arbitrary closed 2-dimensional oriented manifold into R 3 with uniformly bounded conformal class and energy below 12π.

Quantization of the Willmore Energy in Riemannian Manifolds

“…As already mentioned, the first quantization result for the Willmore energy was obtained by Bernard-Rivière [4] for Willmore surfaces with bounded conformal structures and immersed in Euclidean ambient spaces. A first generalisation of [4] was established by Laurain-Rivière [26] in the case of Willmore immersions with degenerating conformal classes, still with values into R n . The crux of the proof of the quantization of energy is to obtain the no-neck energy property, once a suitable decomposition of the domain is performed.…”

Foliation by Area-constrained Willmore Spheres Near a Nondegenerate Critical Point of the Scalar Curvature

“…The existence of a minimizer for every genus was settled by Bauer-Kuwert [2], Kusner [14] and Rivière [33,34] who also developed an independent regularity theory holding more generally for stationary points of W . We also wish to mention the work by Kuwert-Schätzle [16] on the Willmore flow and by Bernard-Rivière [3] and Laurain-Rivière [22] on bubbling and energy-identities phenomena.…”

Minimal Bubbling for Willmore Surfaces

“…In the specific case of limits of Willmore immersions, the punctured disk described in theorem 1.4 is in fact the limit of simply connected disks, on which the first residue is null (see remark 1.1 in [12]). Since away from the concentration point ∇ H k − 3π n k ∇ H k + ∇ ⊥ n k × H k converges, the first residue around branch points of limit Willmore surfaces is always null.…”

“…If the concentration point is not branched, we refer the reader to the concluding remark of P. Laurain and T. Rivière's [12] (found just before the appendix) which states that the energy is then at least β 1 + 12π, where β 1 is the infimum of the Willmore energy of Willmore tori. We would then be above our 12π ceiling, which concludes the proof.…”

“…To do that we will perform a blow-up at the origin at scale µ 3 . This concentration scale has been determined the classical way (see the bubble tree extraction procedure in [12] or [4]) by computing ∇ n Ψµ L ∞ (S 2 ) . Since these computations do not by themselves further the understanding of the bubbling phenomenons, they are omitted.…”