Index of Embedded Networks in the Sphere

TL;DR

This paper computes the Morse index and nullity for stationary embedded networks in spheres, revealing their spectral properties and relating the index to smaller networks and a new Dirichlet-to-Neumann map.

Contribution

It introduces a method to relate the index of complex networks to smaller ones and defines a Dirichlet-to-Neumann map for these networks.

Findings

The index and nullity are related to small networks and a new Dirichlet-to-Neumann map.

For stationary triple junction networks in S^2, only one eigenvalue is less than zero.

Eigenfunctions corresponding to eigenvalues zero are generated by sphere rotations.

Abstract

In this paper, we will compute the Morse index and nullity for the stationary embedded networks in spheres. The key theorem in the computation is that the index (and nullity) for the whole network is related to the index (and nullity) of small networks and the Dirichlet-to-Neumann map defined in this paper. Finally, we will show that for all stationary triple junction networks in $\mathbb{S}^2$, there is only one eigenvalue (without multiplicity) $-1$, which is less than 0, and the corresponding eigenfunctions are locally constant. Besides, the multiplicity of eigenvalues 0 is 3 for these networks, and their eigenfunctions are generated by the rotations on the sphere.