Large Genus Limit of Energy Minimizing Compact Minimal Surfaces in the 3-Sphere

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We aim at finding a pathway towards an implicit function theorem argument to confirm the Kusner conjecture for compact surfaces of high genus, which states that the (simplest) Lawson surfaces minimize the Willmore energy among compact surfaces of genus g. Moreover, we aim at constructing generating functions for the area (resp. the Willmore energy) of the Lawson surfaces. Our investigations include1) classifying all possible large genus limits of Willmore minimizers of genus g. We conjecture the only limit surface to be two intersecting 2-spheres at right angle. The precise asymptotic will help to find appropriate Sobolev spaces to define closeness of immersions of high genera;2) generalizing our DPW deformation methods to CMC surfaces, Willmore surfaces, and to other initial conditions. In particular, we aim at reconstructing the Kapouleas surfaces via DPW and show that their Willmore energy is higher than the ones of the Lawson surfaces with the same genus;3) showing that all symmetric Willmore immersions have a DPW potential of a particular Ansatz type.As a corollary we obtain that the Lawson surfaces \xi_{1,g} are the only symmetric surfaces close to the two intersecting 2-spheres;4) compute further terms in the area expansion of Lawson surfaces. Find recursive formulas for its coefficients; 5) generalization of the results to the whole Lawson family. In particular, computation of the area of the Lawson surfaces in terms as a function of their genus.