Áurea Casinhas Quintino

arXiv2019

Polynomial conserved quantities for constrained Willmore surfaces

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.

math.DG↗

arXiv2013

Constrained Willmore Surfaces: Symmetries of a Moebius Invariant Integrable System

This work is dedicated to the study of the Moebius invariant class of constrained Willmore surfaces and its symmetries. We define a spectral deformation by the action of a loop of flat metric connections; Baecklund transformations, by applying a dressing action; and, in 4-space, Darboux transformations, based on the solution of a Riccati equation. We establish a permutability between spectral deformation and Baecklund transformation and prove that non-trivial Darboux transformation of constrained Willmore surfaces in 4-space can be obtained as a particular case of Baecklund transformation. All these transformations corresponding to the zero multiplier preserve the class of Willmore surfaces. We verify that, for special choices of parameters, both spectral deformation and Baecklund transformation preserve the class of constrained Willmore surfaces admitting a conserved quantity, and, in particular, the class of CMC surfaces in 3-dimensional space-form.

math.DG↗