Morse index of constant mean curvature surfaces and discrete constant mean curvature surfaces

常平均曲率面和离散常平均曲率面的莫尔斯指数

基本信息

批准号：
12640070
负责人：
ROSSMAN Wayne
金额：
$ 2.05万
依托单位：
KOBE UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640070/
关键词：
minimal surface constant mean curvature surfaces Euclidean 3-space hyperbolic 3-space Morse index discrete surfaces integrable systems spherical 3-space 平均曲率一定モース指数 discrete spectrum Weyl asymptotic formula 完備ではない多様体平均曲率が一定

项目摘要

(1) Working with Konrad Polthier, we considered a variational approach for defining discrete minimal surfaces, and established a variational approach for defining discrete constant mean curvature surfaces. We constructed discrete catenoids and helicoids and Delaunay surfaces, and we completely classified the case of catenoids. Furthermore, we computed the discrete Morse index of discrete minimal surfaces and used these computations to examine the Morse index of smooth minimal surfaces. (2) The head investigator proved that Wente tori (these are genus 1 compact constant mean curvature surfaces in R^3) have Morse index at least 7, and also found a lower bound for the Morse index of Wente tori that grows with the spectral genus of the surface. Furthermore, working with Lima and Sousa Neto, we improved the lower bound estimate of 7 to 8. (3) Working with Lima and Berard, we determined the growth rate of the Morse index on complete noncompact constant mean curvature surfaces. (4) Working with Thayer and Wohlgemuth, we constructed many examples of doubly-periodic minimal surfaces in R^3. (5) Working with Umehara and Yamada, we classified all constant mean curvature 1 surfaces in hyperbolic 3-space H^3 that have total curvature at most 8π. (6) Working with Umehara and Yamada and Kokubu, we have started a project to study the nature of singular points on flat surfaces in H^3. (7) Working with Schmitt and Kilian, we have started a project to study constant mean curvature surfaces of genus 0 with three asymptotically Delaunay ends in R^3 and H^3 and the 3-dimensional spherical space S^3.

（1）与Konrad Polthier合作，我们考虑了一种定义离散最小表面的多种方法，并建立了一种定义离散恒定均值曲率表面的多种方法。我们构建了离散的链球菌，螺旋叶和Delaunay表面，并完全分类了相关的情况。此外，我们计算了离散最小表面的离散摩尔斯指数，并使用这些计算来检查平滑最小表面的摩尔斯索引。（2）头部研究员证明了wente tori（这些是r^3中的属1紧凑型恒定平均曲率表面）至少具有7个摩尔斯指数，并且发现Wente Tori的Morse指数下降，随着表面的光谱属而生长。此外，我们与利马和苏萨·内多（Sousa Neto）合作，提高了7至8的较低估计值。（3）与利马和贝拉德（Lima）和贝拉德（Berard）合作，我们确定了摩尔斯（Morse）指数的增长率在完全非稳定的恒定恒定平均曲率表面上。（4）与Thayer和Wohlgemuth合作，我们在R^3中构建了许多双周期最小表面的例子。（5）与Umehara和Yamada一起工作，我们将所有恒定平均曲率1表面分类为双曲线3空间H^3，最多具有8π。（6）与Umehara，Yamada和Kokubu合作，我们已经开始了一个项目，以研究H^3中平坦表面上奇异点的性质。（7）与Schmitt和Kilian合作，我们已经开始了一个项目，研究了属0的恒定平均曲率表面，三个不对称的Delaunay以R^3和H^3结束，以及3维球形空间S^3。