3-circle Theorem for Willmore surfaces II--degeneration of the complex structure

2024/11/10 by Yuxiang Li, Li, Yuxiang, Yin Hao +3
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2411.06453

openalex publication_date 2024/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the compactness of Willmore surfaces without assuming the convergence of the induced complex structures. In particular, we compute the energy loss in the neck in terms of the residue and we prove that the limit of the image of the Gauss map is a geodesic in the Grassmannian G(2,n) whose length can also be computed in terms of the residue. Moreover, we provide a family of explicit Willmore surfaces in \R³ that illustrate the denegeration phenomenon involved in the above results.

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