arXiv subjects

Xuanpu Liang

Publications and source records attributed to Xuanpu Liang.

3 recordsLinked to original sources

arXiv2026

Componentwise Geometry and Monodromy of Generalized Lam\'e Equations

We develop a componentwise geometric and monodromy theory for the one-support generalized Lam\'e equation on an elliptic curve, with singularities at $0$ and $\pm p$. Its log-free curve decomposes canonically into irreducible even and non-even components, the former being governed by elliptic Painlev\'e~VI. For the non-even component, we construct the hyperelliptic spectral curve, Baker--Akhiezer functions, and addition map, and prove that \[ \deg \sigma_{n,p}^{(1)}=n(n+1). \] After quotienting by the involution $T\mapsto -T$, we identify the non-even spectral curve with the classical Lam\'e spectral curve of weight $n$, compatibly with the addition map and the rational function $\kappa$. This identification is realized by \[ \widetilde B=T^2-n(n+1)\wp(p), \] and associates every non-even generalized Lam\'e equation with a unique classical Lam\'e equation on the same elliptic curve having equivalent period monodromy. Fixing $\widetilde B$ yields an isomonodromic deformation with $\tau$ fixed. Together with the Painlev\'e-VI deformation on the even component, it gives a componentwise interpretation of the collision $p\to0$, and yields a finite descent on the admissible completely reducible locus. The classical spectral, finite-gap, finite-monodromy, and curvature theories consequently transfer to the non-even component. Finally, within the symmetric family $\left(n_0,n_1,n_2,n_3,\frac12,\frac12\right),$ the one-support case forms an affine genus-zero hierarchy, whereas for $\left(1,1,0,0,\frac12,\frac12\right)$, the non-even normalization is generically elliptic and becomes rational on the discriminant locus, while the full compactified log-free curve retains arithmetic genus two. This first genus jump marks the boundary of the affine theory and motivates a genus-dependent componentwise geometry.

math.DG↗

arXiv2025

Even Cone Spherical Metrics: Blow-Up at a Cone Singularity

We study families of spherical metrics on the flat torus $E_{\tau}$ $=$ $\mathbb{C}/\Lambda_{\tau}$ with blow-up behavior at prescribed conical singularities at $0$ and $\pm p$, where the cone angle at $0$ is $6\pi$, and at $\pm p$ is $4\pi$. We prove that the existence of such a necessarily unique, even family of spherical metrics is completely determined by the geometry of the torus: such a family exists if and only if\textbf{ }the Green function $G(z;\tau)$ admits a pair of nontrivial critical points $\pm a$. In this case, the cone point $p$ must equal $a$, and the corresponding monodromy data is $\left( 2r,2s\right) $, where $a=r+s\tau.$ An explicit transformation relating this family to the one with a single conical singularity of angle $6\pi$ at the origin is established in Theorem 1.4. A rigidity result for rhombic tori is proved in Theorem 1.5.

math.DG↗

arXiv2025

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

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.

math.CA↗