Constrained Willmore Surfaces Symmetries Of A Meobius Invariant Integrable System Urea Casinhas Quintino by Áurea Casinhas Quintino 9781108885478, 1108885470 instant download after payment.

"This work is dedicated to the study of the Mèobius invariant class of constrained Willmore surfaces and its symmetries. Characterized by the perturbed harmonicity of the mean curvature sphere congruence, a generalization of the well-developed integrable systems theory of harmonic maps emerges. The starting point is a zero-curvature characterization, due to Burstall-Calderbank, which we derive from the underlying variational problem. Constrained Willmore surfaces come equipped with a family of flat metric connections. We then define a spectral deformation, by the action of a loop of flat metric connections; Bèacklund transformations, defined by the application of a version of the Terng-Uhlenbeck dressing action by simple factors; and, in 4-space, Darboux transformations, based on the solution of a Riccati equation, generalizing the transformation of Willmore surfaces presented in the quaternionic setting by Burstall-Ferus-Leschke-Pedit-Pinkall. We establish a permutability between spectral deformation and Bèacklund transformation and prove that non-trivial Darboux transformation of constrained Willmore surfaces in 4-space can be obtained as a particular case of Bèacklund transformation. All these transformations corresponding to the zero Lagrange multiplier preserve the class of Willmore surfaces. We verify that both spectral deformation and Bèacklund transformation preserve the class of constrained Willmore surfaces admitting a conserved quantity, defining, in particular, transformations within the class of constant mean curvature surfaces in 3-dimensional space-forms, with, furthermore, preservation of both the space-form and the mean curvature, in the latter case. Constrained Willmore transformation proves to be unifying to the rich transformation theory of CMC surfaces in 3-space"--