Index of embedded networks in the sphere

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In this paper, we will compute the Morse index and nullity of some embedded stationary geodesic networks in the sphere. 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 S2, 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.