Complete Willmore surfaces in H3 with bounded energy: boundary regularity and bubbling

2012/04/22 by Spyros Alexakis, Rafe Mazzeo, Alexakis, Spyros +1 · 3 citations
Computer Science · Mathematics · #35J70 #53A10 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations #math.AP #math.DG #msc:35J70 #msc:53A10

paper · pdf · doi:10.48550/arxiv.1204.4955

Substantial revisions: basic energy hypothesis strengthened; the result now covers Willmore surfaces

openalex publication_date 2012/04/22 · arxiv created 2013/07/29 · arxiv updated 2016/11/26 · openalex created_date 2022/09/05 · openalex updated_date 2026/07/28

Abstract

We study various aspects related to boundary regularity of complete properly embedded Willmore surfaces in H3, particularly those related to assumptions on boundedness or smallness of a certain weighted version of the Willmore energy. We prove, in particular, that small energy controls C1 boundary regularity. We examine the possible lack of C1 convergence for sequences of surfaces with bounded Willmore energy and find that the mechanism responsible for this is a bubbling phenomenon, where energy escapes to infinity.

Citations

• Topics in conformally compact Einstein metrics
• Weak limit of an immersed surface sequence with bounded Willmore functional
• Existence an regularity of constant mean curvature hypersurfaces in hyperbolic space
• Multiple-layer solutions to the Allen-Cahn equation on hyperbolic space

Cited by

• Asymptotic behavior of geodesics in conformally compact manifolds
• Compactness and Bubbles Analysis for 1/2-harmonic Maps
• Existence of Generalized Totally Umbilic 2-Spheres in Perturbed 3-Spheres

Related

• Higher Regularity of Weak Limits of Willmore Immersions I
• Energy Quantization for Willmore Surfaces and Applications
• Energy quantization for constrained Willmore surfaces
• Higher Regularity of Weak Limits of Willmore Immersions II
• The Willmore Energy Landscape of Spheres and Avoidable Singularities of the Willmore Flow