Monodromy Equivalence for Lam\'{e}-type Equations I: Finite-gap Structures and Cone Spherical Metrics

Abstract

Motivated by the finite-gap structure of the classical Lam\'{e} equation (1.2) and its central role in mathematical physics, generalized Lam\'{e}-type equations (1.12) are investigated. For the fundamental case $n=1$, a monodromy equivalence between the classical Lam\'{e} equation (1.18) and the generalized Lam\'{e}-type equation (1.19) is established. Two main applications are obtained: (i) the finite-gap structure of \ (1.19) is derived, together with a complete classification of the spectral curves $\sigma_{1}$ and $\sigma_{2}$ for $\tau\in i\mathbb{R}_{>0}$; and (ii) the monodromy equivalence is applied to the construction of cone spherical metrics with three large conical singularities, each with cone angle exceeding $2\pi$. A family of such metrics is shown to exhibits a blow-up configuration, which is described explicitly in terms of the monodromy data.