Spectral properties of reducible conical metrics

Abstract

We show that the monodromy of a spherical conical metric g is reducible if and only if the metric g has a real-valued eigenfunction with eigenvalue 2 for the holomorphic extension ${\mathrm{\Delta }}_g^{\mathrm{Hol}}$ of the associated Laplace–Beltrami operator. Such an eigenfunction produces a meromorphic vector field, which is then related to the developing maps of the conical metric. We also give a lower bound of the first nonzero eigenvalue of ${\mathrm{\Delta }}_g^{\mathrm{Hol}}$, together with a complete classification of the dimension of the space of real-valued 2-eigenfunctions for ${\mathrm{\Delta }}_g^{\mathrm{Hol}}$ depending on the monodromy of the metric g. This paper can be seen as a new connection between the complex analysis method and the PDE approach in the study of spherical conical metrics.