Polynomial conserved quantities for constrained Willmore surfaces

2015/07/05 by Áurea Casinhas Quintino, Quintino, Áurea Casinhas, S. D. Santos +1
Mathematics · Physics and Astronomy · #53A05 #53A10 #53A30 #53C43 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1507.01253

openalex publication_date 2015/07/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We define a hierarchy of special classes of constrained Willmore surfaces by means of the existence of a polynomial conserved quantity of some type, filtered by an integer. Type 1 with parallel top term characterises parallel mean curvature surfaces and, in codimension 1, type 1 characterises constant mean curvature surfaces. We show that this hierarchy is preserved under both spectral deformation and Baecklund transformation, for special choices of parameters, defining, in particular, transformations of constant mean curvature surfaces into new ones, with preservation of the mean curvature, in the latter case.

Citations

• Conformal submanifold geometry I-III

Related

• Constrained Willmore Surfaces: Symmetries of a Moebius Invariant Integrable System
• Willmore surfaces in spheres via loop groups: a survey
• Harmonic map methods for Willmore surfaces
• Links between the integrable systems of CMC surfaces, isothermic surfaces and constrained Willmore surfaces
• Analysis aspects of Willmore surfaces