Critical weak immersed surfaces within sub-manifolds of the Teichmüller space

We prove that the critical points of various energies such as the area, the Willmore energy, the frame energy for tori, etc., among possibly branched immersions constrained to evolve within a smooth sub-manifold of the Teichmüller space satisfy the corresponding constrained Euler Lagrange equation. We deduce that critical points of the Willmore energy or the frame energy for tori are smooth analytic surfaces, away possibly from isolated branched points, under the condition that either the genus is at most 2 or if the sub-manifold does not intersect the subspace of hyper-elliptic points. Using a compactness result from [19] we can conclude that each closed sub-manifold of the Teichmüller space, including points, under the previous assumptions, possess a possibly branched smooth Willmore minimizer satisfying the conformal-constrained Willmore equation.