The Morse index of a triply periodic minimal surface
The Morse index of a triply periodic minimal surface
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三周期极小曲面的莫尔斯指数
DOI：
10.1016/j.difgeo.2018.01.006
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发表时间：
2018
影响因子：
0.5
通讯作者：
Shoda Toshihiro
中科院分区：
文献类型：
--
作者：
Ejiri Norio;Shoda Toshihiro
在之前的工作中，第一作者建立了一个计算R n中n周期极小曲面的莫尔斯指数和零度的算法。实际上，莫尔斯指标可以转化为真实的对称矩阵的负特征值个数，零度可以转化为厄米特矩阵的零特征值个数.这两个关键矩阵由极小曲面上的第二类交换微分的周期组成，而Hermitian矩阵的签名给出了极小曲面的一个新的不变量。另一方面，在物理、化学和晶体学等方面研究了R3中H族、rPD族、tP族、tD族和tCLP族三重周期极小曲面。在本文中，我们首先明确地确定了这五个家庭的两个关键矩阵。作为应用，我们通过数值计算，计算了这五个族的莫尔斯指数、零数和签名。
In the previous work, the first author established an algorithm to compute the Morse index and the nullity of an n-periodic minimal surface in R n. In fact, the Morse index can be translated into the number of negative eigenvalues of a real symmetric matrix and the nullity can be translated into the number of zero-eigenvalue of a Hermitian matrix. The two key matrices consist of periods of the abelian differentials of the second kind on a minimal surface, and the signature of the Hermitian matrix gives a new invariant of a minimal surface. On the other hand, H family, rPD family, tP family, tD family, and tCLP family of triply periodic minimal surfaces in R 3 have been studied in physics, chemistry, and crystallography. In this paper, we first determine the two key matrices for the five families explicitly. As its applications, by numerical arguments, we compute the Morse indices, nullities, and signatures for the five families.