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

2016/06/30 by Paul Laurain, Tristan Rivière
Mathematics · #Advanced Harmonic Analysis Research #Boundary (topology) #Curvature #Geometric Analysis and Curvature Flows #Geometry #Mathematical Analysis and Transform Methods #Mathematical analysis #Mathematics #Mean curvature #Moduli #Moduli space #Physics #Pure mathematics #Quantization (signal processing) #Quantum mechanics #Willmore energy #math.AP #math.DG

paper · pdf · doi:10.1215/00127094-2018-0010

published as Duke Math. J. 167, no. 11 (2018), 2073-2124

arxiv created 2016/09/28 · openalex publication_date 2018/06/07 · arxiv updated 2018/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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 possible loss of energy in collar regions. Thanks to this residue, we also establish the compactness (modulo the action of the Möbius group of conformal transformations of R3∪∞) of the space of Willmore immersions of any arbitrary closed 2-dimensional oriented manifold into R3 with uniformly bounded conformal class and energy below 12π.

Citations

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