Analysis aspects of Willmore surfaces

• 成果类型: Article
• 署名作者: Riviere, Tristan
• 署名单位: Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-008-0129-7
• 发表日期: 2008
• 页码: 1-45
• 关键词: theorem; FLOW; invariant; EXISTENCE

摘要

A new formulation for the Euler-Lagrange equation of the Willmore functional for immersed surfaces in R-m is given as a nonlinear elliptic equation in divergence form, with non-linearities comprising only Jacobians. Letting (H) over right arrow be the mean curvature vector of the surface, our new formulation reads L (H) over right arrow = 0, where L is a well-defined locally invertible self-adjoint elliptic operator. Several consequences are studied. In particular, the long standing open problem asking for a meaning to the Willmore Euler-Lagrange equation for immersions having only L-2-bounded second fundamental form is now solved. The regularity of weak Willmore immersions with L-2-bounded second fundamental form is also established. Its proof relies on the discovery of conservation laws which are preserved under weak convergence. A weak compactness result for Willmore surfaces with energy less than 8 pi (the Li-Yau condition ensuring the surface is embedded) is proved, via a point removability result established for Wilmore surfaces in R-m, thereby extending to arbitrary codimension the main result in [KS3]. Finally, from this point-removability result, the strong compactness of Willmore tori below the energy level 8 pi is proved both in dimension 3 (this had already been settled in [KS3]) and in dimension 4.