2013/07/20 by Tristan Rivière, Rivière, Tristan
Mathematics · #35J35 #49Q10 #53A05 #53A30 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:35J35 #msc:49Q10 #msc:53A05 #msc:53A30
paper · pdf · doi:10.48550/arxiv.1307.5406
arxiv created 2013/07/20 · arxiv updated 2013/07/23
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 a previous work of the author, we can conclude that each closed sub-manifold of the Teichmüller space, including points, under the previous assumptions, posses a possibly branched smooth Willmore minimizer satisfying the conformal-constrained Willmore equation.