The Morse index of a triply periodic minimal surface

TL;DR

This paper develops an algorithm to compute the Morse index and nullity of triply periodic minimal surfaces, explicitly determines key matrices for five families, and numerically computes their indices, nullities, and signatures.

Contribution

It introduces a method to explicitly compute key matrices and Morse indices for specific triply periodic minimal surface families, advancing understanding of their stability properties.

Findings

Explicit matrices for five minimal surface families

Numerical Morse indices and nullities computed

Signature invariants introduced for minimal surfaces

Abstract

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 $\mathbb{R}^n$. 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 $\mathbb{R}^3$ 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.