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

我们的目的是找到一条通往隐函数定理论证的途径来证实高亏格紧致曲面的Kusner猜想，即(最简单的)Lawson曲面最小化g亏格紧致曲面中的Willmore能量。此外，我们还构造了面积的母函数。威尔莫尔能量)。我们的研究包括：1)对亏格g的Willmore极小子的所有可能的大亏格极限进行分类，我们猜想唯一的极限曲面是两个相交的直角2-球面。精确的渐近性将有助于找到合适的Sobolev空间来定义高亏格的浸入的封闭性；2)将我们的DPW变形方法推广到CMC曲面、Willmore曲面和其他初始条件。特别地，我们利用DPW来重构Kapouleas曲面，并且证明了它们的Willmore能量高于具有相同亏格的Lawson曲面的Willmore能量；3)证明了所有对称的Willmore浸没都具有特定Anatz型的DPW势.作为推论，我们得到了Lawson曲面是唯一靠近两个相交的2-球面的对称曲面；4)计算了Lawson曲面的面积扩展项.求其系数的递推公式；5)将结果推广到整个Lawson族。特别地，计算Lawson曲面作为亏格的函数的面积。

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.