Index of Embedded Networks in the Sphere

Tue, 01 Oct 2024 11:51:01 +0800

Main results

This paper develops a spectral decomposition method for stationary geodesic networks and applies it to every embedded closed triple-junction network in the round two-sphere.

Exact index

The Morse index is determined by the complementary regions

If the network cuts $\mathbb S^2$ into $F$ regions, then its Morse index is exactly $F-1$. All negative modes have eigenvalue $-1$ and are locally constant along the edges.

Exact nullity

The only Jacobi fields come from rotations

Every embedded closed stationary triple-junction network has nullity three, with the zero modes generated precisely by infinitesimal rotations of $\mathbb S^2$.

Decomposition formula

Large networks reduce to smaller pieces

For an arbitrary partitioned network, the index and nullity split into contributions from the subnetworks, a finite-dimensional Dirichlet-to-Neumann operator, and explicit matching spaces at the cut points.

Method

The Jacobi operator is $L=\Delta+1$, acting on edgewise functions satisfying the compatibility conditions imposed by geometric variations at junctions. After cutting the network, solutions of the Jacobi equation on each piece are encoded by a self-adjoint Dirichlet-to-Neumann map. The resulting index formula turns the global spectral calculation into finite-dimensional linear algebra and, together with the classification of spherical triple-junction networks, yields the exact spectrum at and below zero.

Geodesic | Gaoming Wang

Index of Embedded Networks in the Sphere

Main results

The Morse index is determined by the complementary regions

The only Jacobi fields come from rotations

Large networks reduce to smaller pieces

Method