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Index of Embedded Networks in the Sphere

Gaoming Wang

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.

Index of Embedded Networks in the Sphere

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 , there is only one eigenvalue (without multiplicity) , 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.

Paper Structure

This paper contains 12 sections, 16 theorems, 130 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Curves and networks in $\mathbb S^2$
  4. Some useful function spaces
  5. Examples of eigenfunctions
  6. The Spectrum on networks and the Dirichlet-to-Neumann map
  7. Eigenvalues on networks
  8. The Dirichlet-to-Neumann map
  9. Index Theorem for networks
  10. Dirichlet-to-Neumann map with respect to a partition
  11. Index and nullity of the stationary triple junction networks on spheres
  12. Computation of Dirichlet-to-Neumann maps

Key Result

Theorem 1.1

The Morse index of all embedded closed stationary triple junction networks in $\mathbb{S}^2$ is $F-1$ where $F$ is the number of regions on the sphere cut by this network. The corresponding eigenfunctions are all locally constant (explained in Theorem thm_index_and_nullity_of_triple_junction_network

Figures (7)

  • Figure 1: A regular spherical tetrahedron
  • Figure 2: A part of a triple junction network
  • Figure 3: Partition of one-skeleton of the regular spherical polyhedron
  • Figure 4: Partition of prism-type networks
  • Figure 5: Partition of 4-4 type network
  • ...and 2 more figures

Theorems & Definitions (67)

  • Theorem 1.1: Index and nullity of stationary triple junction networks
  • Theorem 1.2: Index and nullity theorem
  • Remark 1.3
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 57 more