Local Palais-Smale Sequences for the Willmore Functional

2009/04/02 by Yann Bernard, Bernard, Yann, Tristan Rivière +2 · 2 citations
Economics, Econometrics and Finance · Mathematics · #35Jxx #53A05 #53A30 #53C42 #58E15 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Stochastic processes and financial applications #math.AP #math.DG #msc:35Jxx #msc:53A05 #msc:53A30 #msc:53C42 #msc:58E15

paper · pdf · doi:10.48550/arxiv.0904.0360

35 pages

arxiv created 2009/04/02 · openalex publication_date 2009/04/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Using the reformulation in divergence form of the Euler-Lagrange equation for the Willmore functional as it was developed in "Analysis of the Willmore Functional" by T. Riviere (Invent. Math. 174), we study the limit of a local Palais-Smale sequence of weak Willmore immersions with locally square-integrable second fundamental form. We show that the limit immersion is smooth and that it satisfies the conformal Willmore equation: it is a critical point of the Willmore functional restricted to infinitesimal conformal variations.

Citations

• A theorem on geometric rigidity and the derivation of nonlinear plate theory from three‐dimensional elasticity
• A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces

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