5 - The Euler–Lagrange ConstrainedWillmore Surface Equation

Summary

In this chapter, we introduce constrained Willmore surfaces, the generalization of Willmore surfaces that arises when we consider critical points of the Willmore functional with respect to infinitesimally conformal variations, rather than with respect to all variations. Constrained Willmore surfaces in space-forms constitute a Möbius invariant class of surfaces with strong links to the theory of integrable systems, which we shall explore throughout this work. The introduction of a constraint in the variational problem equips surfaces with Lagrange multipliers, as first proven by Burstall–Pedit–Pinkall, in a manifestly conformally invariant characterization of constrained Willmore surfaces in space-forms in terms of the Hopf differential and the Schwarzian derivative, which we deduce in this chapter. The class of constrained Willmore surfaces was later given yet another manifestly conformally invariant characterization, free of holomorphic charts, by Burstall–Calderbank, which we derive in this chapter from the variational problem.