2018/01/31 by Norio Ejiri, Ejiri, Norio, Toshihiro Shoda +1
Mathematics · #53C42 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Graph theory and applications #Primary 53A10 #Secondary 49Q05 #math.DG #msc:49Q05 #msc:53A10 #msc:53C42
paper · pdf · doi:10.48550/arxiv.1801.10388
This paper is the original version. The improved version is to appear in Differential Geometry and its Applications
arxiv created 2018/01/31 · openalex publication_date 2018/01/31 · arxiv updated 2018/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ℝn. 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 ℝ3 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.