Application of integrable systems methods to surfaces with particular variational properties

1)在与U. Hertrich-Jeromin， S. Santos和F. Burstall的联合研究项目中，获得了三维空间形式下离散常平均曲率曲面的合适定义。这些三维空间形式包括欧几里得三维、球面三维和双曲三维。结果表明，这个新定义与欧几里得情况下的旧定义相匹配，而这个新定义在双曲情况下是新的。利用这一定义，研究了离散的Delaunay曲面及其离散的Darboux和Backlund变换。这项研究的一个重要工具是守恒量的概念。光滑表面的情况是由S. Santos和F. Burstall提出的，而离散情况是由U. Hertrich-Jeromin和我提出的。2)在与博士研究生N. Sultana的联合研究项目中，研究了球面三维空间中常平均曲率旋转曲面的稳定性和Morse指数。因为这样一个曲面的轴是一个闭环，这些曲面可以变成闭合环面，然后它们就会有有限的指标。结果表明，所有这些表面都是不稳定的，它们的索引值都至少为5，并且（取决于表面的选择）索引值可以任意大。索引是相关雅可比算子的负特征值的个数。3)在与M. Kokubu， M. Umehara和K. Yamada合作的项目的延续中，研究了双曲三维空间中具有恒定高斯曲率0的曲面（可以有奇点的平坦面）。特别是，今年的研究表明，这样的曲面的焦散量可以具有由摆线描述的渐近行为的端点。

The following results were obtained:1) In a joint research project with U. Hertrich-Jeromin, S. Santos and F. Burstall, a suitable definition for discrete constant mean curvature surfaces in 3 dimensional space forms was obtained. Those 3 dimensional space forms consist of Euclidean 3-space, spherical 3-space and hyperbolic 3-space. It was shown that this new definition matches the old definition that is known for the Euclidean case, and this definition is new in the hyperbolic case. Using this definition, discrete Delaunay surfaces were studied, along with their discrete Darboux and Backlund transformations. An important tool in this research was the notion of conserved quantities. The case of smooth surfaces was developed by S. Santos and F. Burstall, while the discrete case was developed by U. Hertrich-Jeromin and myself.2) In a joint research project with my Ph.D. graduate student N. Sultana, the stability and Morse index of constant mean curvature surfaces of revolution in spherical 3-space was studied. Because the axis of such a surface is a closed loop, these surfaces can become close tori, and then they will have finite index. It was shown that all such surfaces are unstable, and that they all have index at least 5, and (depending on the choice of surface) the index can be arbitrarily large. The index is the number of negative eigenvalues of the associated Jacobi operator.3) In a continuation of a project with M. Kokubu, M. Umehara and K. Yamada, surfaces with constant Gauss curvature 0 in hyperbolic 3-space (flat fronts, which can have singularities) were studied. In particular, this year, it was shown that the caustics of such surfaces can have ends with asymptotic behavior described by cycloids.