Index of embedded networks in the sphere (Q6581808)

This paper considers networks, namely one-dimensional singular versions of surfaces. As the main result, the Morse index and nullity of stationary triple junction networks in the sphere \(\mathbb{S}^2\) are computed (Theorem 1.1 and in a more formal version Theorem 5.1). The key theorem in the computation is that the index (and nullity) for the whole network is related to the index (and nullity) of smaller networks and the Dirichlet-to-Neumann map defined in the paper (Theorem 1.2). \N\NThe paper is organized into five sections as follows: \N\begin{itemize}\N\item Introduction;\N\item Preliminaries (curves and networks in \(\mathbb{S}^2\), some useful function spaces, examples of eigenfunctions);\N\item The spectrum on networks and the Dirichlet-to-Neumann map (eigenvalues on networks);\N\item Index theorem for networks (Dirichlet-to-Neumann map with respect to a partition);\N\item Index and nullity of the stationary triple junction networks on spheres. \N\end{itemize} A useful appendix ``Computation of Dirichlet-to-Neumann maps'' is given at the end of the paper. Another paper of the author directly connected to this topic is [Calc. Var. Partial Differ. Equ. 61, No. 4, Paper No. 138, 26 p. (2022; Zbl 1502.53014)].