7 - Constrained Willmore Surfaces with a Conserved Quantity

Summary

In this chapter, we introduce the concept of conserved quantity of a constrained Willmore surface in a space-form. In codimension 1, the existence of a conserved quantity for a constrained Willmore surface characterizes the constancy of the mean curvature of the surface in some space-form. In codimension 2, surfaces with holomorphic mean curvature vector in some space-form are examples of constrained Willmore surfaces admitting a conserved quantity. Both constrained Willmore spectral deformation and Bäcklund transformation preserve the existence of a conserved quantity of a constrained Willmore surface, 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, as we shall prove.