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Ting-Jung Kuo

Publications and source records attributed to Ting-Jung Kuo.

At least 19 recordsLinked to original sources
arXiv2026

Componentwise Geometry and Monodromy of Generalized Lamé Equations

We develop a componentwise geometric and monodromy theory for the one-support generalized Lamé 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 Painlevé~VI. For the non-even component, we construct the hyperelliptic spectral curve, Baker--Akhiezer functions, and addition map, and prove that \[ °σ_{n,p}^{(1)}=n(n+1). \] After quotienting by the involution \(T\mapsto -T\), we identify the non-even spectral curve with the classical Lamé spectral curve of weight \(n\), compatibly with the addition map and the rational function \(κ\). This identification is realized by \[ \widetilde B=T^2-n(n+1)\wp(p), \] and associates every non-even generalized Lamé equation with a unique classical Lamé equation on the same elliptic curve having equivalent period monodromy. Fixing \(\widetilde B\) yields an isomonodromic deformation with \(τ\) fixed. Together with the Painlevé-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 $E_τ$ $=$ $\mathbb{C}/Λ_τ$ with blow-up behavior at prescribed conical singularities at $0$ and $\pm p$, where the cone angle at $0$ is $6π$, and at $\pm p$ is $4π$. 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 $G(z;τ)$ 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 $a=r+sτ.$ An explicit transformation relating this family to the one with a single conical singularity of angle $6π$ 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 Lamé-type Equations I: Finite-gap Structures and Cone Spherical Metrics

Motivated by the finite-gap structure of the classical Lamé equation (1.2) and its central role in mathematical physics, generalized Lamé-type equations (1.12) are investigated. For the fundamental case $n=1$, a monodromy equivalence between the classical Lamé equation (1.18) and the generalized Lamé-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 $σ_{1}$ and $σ_{2}$ for $τ\in i\mathbb{R}_{>0}$; and (ii) the monodromy equivalence is applied to the construction of cone spherical metrics with three large conical singularities, each with cone angle exceeding $2π$. 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↗
arXiv2025

Sovability of curvature equations with multiple singular sources on torus via Painleve VI equations

We study the curvature equation with multiple singular sources on a torus \[Δu+e^{u}=8π\sum_{k=0}^{3}n_{k}δ_{\frac{ω_{k}}{2}}% +4π\left( δ_{p}+δ_{-p}\right) \quad \text{ on }\;E_τ:=\mathbb{C}/(\mathbb Z+\mathbb{Z}τ),\] where $n_k\in\mathbb N$ and $δ_a$ denotes the Dirac measure at $a$. This is known as a critical case for which the apriori estimate does not hold, and the existence of solutions has been a long-standing problem. In this paper, by establishing a deep connection with Painlevé VI equations, we show that the existence of even solutions (i.e. $u(z)=u(-z)$) depends on the location of the singular point $p$, and we give a sharp criterion of $p$ in terms of Painlevé VI equations.

math.AP↗
arXiv2024

Sharp results for spherical metric on flat tori with conical angle 6$π$ at two symmetric points

In this paper, we investigate the following curvature equation: \begin{equation} Δu+e^{u}=8π(δ_{0}+δ_{\frac{ω_{k}}{2}})\text{ in } E_{τ}\text{, }τ\in \mathbb{H} (0.1) \label{a} \end{equation} Here $E_{τ}$ represents a flat torus and $\frac{ω_{k}}{2}$ is one of the half periods of $E_{τ}$. Our primary objective is to establish a necessary and sufficient criterion for the existence of a non-even family of solutions (see the definition in Section 1). Remarkably, this is equivalent to determining the presence of solutions for the equation with a single conical singularity: \begin{equation*} Δu+e^{u}=8πδ_{0}\text{ in }E_{τ}\text{, }τ\in \mathbb{ H}\text{.} \end{equation*} This study marks the first exploration of the structure of non-even families of solutions to the curvature equation with multiple singular sources in the literature. Building on our findings, we provide a comprehensive analysis of the solution structure for equation (0.1) for all $τ$. This analysis is facilitated by Theorem 1.3, which will play a central role in our exploration of cases involving general parameters in the future, such as: \begin{equation*} Δu+e^{u}=8πn(δ_{0}+δ_{\frac{ω_{k}}{2}})\text{ in } E_{τ},\text{ }n\in \mathbb{N}\text{.} \end{equation*} As an application, we offer explicit descriptions for solutions to equation (0.1) in the context of both rectangle tori and rhombus tori. See Corollary 1.4 as well as Corollary 1.5.

math.AP↗
arXiv2021

Proof of a conjecture of Dahmen and Beukers on counting integral Lamé equations with finite monodromy

In this paper, we prove Dahmen and Beukers' conjecture that the number of integral Lamé equations with index $n$ modulo scalar equivalence with the monodromy group dihedral $D_{N}$ of order $2N$ is given by \[L_{n}(N)=\frac{1}{2}\left( \frac{n(n+1)Ψ(N)}{24}-\left( a_{n}% ϕ(N)+b_{n}ϕ(\tfrac{N}{2}) \right) \right) +\frac{2}% {3}\varepsilon_{n}(N).\] Our main tool is the new pre-modular form $Z_{r,s}^{(n)}(τ)$ of weight $n(n+1)/2$ introduced by Lin and Wang \cite{LW2} and the associated modular form $M_{n,N}(τ):=\prod_{(r,s)}Z_{r,s}^{(n)}(τ)$ of weight $Ψ(N)n(n+1)/{2}$, where the product runs over all $N$-torsion points $(r,s)$ of exact order $N$. We show that this conjecture is equivalent to the precise formula of the vanishing order of $M_{n,N}(τ)$ at infinity: \[v_{\infty}(M_{n,N}(τ))=a_{n}ϕ(N)+b_{n}ϕ( N/2).\] This formula is extremely hard to prove because the explicit expression of $Z_{r,s}^{(n)}(τ)$ is not known for general $n$. Here we succeed to prove it by using certain Painlevé VI equations. Our result also indicates that this conjecture is intimately connected with the problem of counting pole numbers of algebraic solutions of certain Painlevé VI equations. The main results of this paper has been announced in \cite{Lin-CDM}.

math.NT↗
arXiv2020

The geometry of generalized Lame equation, III: One-to-one of the Riemann-Hilbert correspondence

In this paper, the third in a series, we continue to study the generalized Lamé equation H$(n_0,n_1,n_2,n_3;B)$ with the Darboux-Treibich-Verdier potential \begin{equation*} y^{\prime \prime }(z)=\bigg[ \sum_{k=0}^{3}n_{k}(n_{k}+1)\wp(z+\tfrac{ ω_{k}}{2}|τ)+B\bigg] y(z),\quad n_{k}\in \mathbb{Z}_{\geq0} \end{equation*} and a related linear ODE with additional singularities $\pm p$ from the monodromy aspect.We establish the uniqueness of these ODEs with respect to the global monodromy data. Surprisingly, our result shows that the Riemann-Hilbert correspondence from the set \[\{\text{H}(n_0,n_1,n_2,n_3;B)|B\in\mathbb{C}\}\cup \{\text{H}(n_0+2,n_1,n_2,n_3;B) | B\in\mathbb{C}\}\] to the set of group representations $ρ:π_1(E_τ)\to SL(2,\mathbb{C})$ is one-to-one. We emphasize that this result is not trivial at all. There is an example that for $τ=\frac12+i\frac{\sqrt{3}}{2}$, there are $B_1,B_2$ such that the monodromy representations of H$(1,0,0,0;B_1)$ and H$(4,0,0,0;B_2)$ are {\bf the same}, namely the Riemann-Hilbert correspondence from the set \[\{\text{H}(n_0,n_1,n_2,n_3;B)|B\in\mathbb{C}\}\cup \{\text{H}(n_0+3,n_1,n_2,n_3;B) | B\in\mathbb{C}\}\] to the set of group representations is {\bf not} necessarily one-to-one. This example shows that our result is completely different from the classical one concerning linear ODEs defined on $\mathbb{CP}^1$ with finite singularities.

math.CA↗
arXiv2020

Blow up at infinity in the SU(3) Chern-Simons model, part I

We consider non-topological solutions of a nonlinear elliptic system problem derived from the $SU(3)$ Chern-Simons models in $\mathbb{R}^2$. The existence of non-topological solutions even for radial symmetric case has been a long standing open problem. Recently, [Choe, Kim, Lin (2015, 2016)] showed the existence of radial symmetric non-topological solution when the vortex points collapse. However, the arguments in [Choe, Kim, Lin (2015, 2016)] cannot work for an arbitrary configuration of vortex points. In this paper, we develop a new approach by using different scalings for different components of the system to construct a family of non-topological solutions, which blows up at infinity.

math.AP↗
arXiv2019

On the CR analogue of Frankel conjecture and a smooth representative of the first Kohn-Rossi cohomology group

In this note, we first give a criterion of pseudo-Einstein contact forms and then affirm the CR analogue of Frankel conjecture in a closed, spherical, strictly pseudoconvex CR manifold of nonnegative pseudohermitian curvature on the space of smooth representatives of the first Kohn-Rossi cohomology group. Moreover, we obtain the CR Frankel conjecture in a closed, spherical, strictly pseudoconvex CR manifold with the vanishing first Kohn-Rossi cohomology group. In particular, this conjecture holds in a spherical boundary of the Stein manifold.

math.DG↗
arXiv2019

Global existence and convergence for the CR Q-curvature flow in a closed strictly pseudoconvex CR 3-manifold

In this note, we affirm the partial answer to the long open Conjecture which states that any closed embeddable strictly pseudoconvex CR $3$-manifold admits a contact form $θ$ with the vanishing CR $Q$-curvature. More precisely, we deform the contact form according to an CR analogue of $Q$%-curvature flow in a closed strictly pseudoconvex CR $3$-manifold $(M,\ J,[θ_{0}])$ of the vanishing first Chern class $c_{1}(T_{1,0}M)$. Suppose that $M$ is embeddable and the CR Paneitz operator $P_{0}$ is nonnegative with kernel consisting of the CR pluriharmonic functions. We show that the solution of CR $Q$-curvature flow exists for all time and has smoothly asymptotic convergence on $M\times \lbrack 0,\infty ).$\ As a consequence, we are able to affirm the Conjecture in a closed strictly pseudoconvex CR $3$-manifold of the vanishing first Chern class and vanishing torsion.

math.DG↗
arXiv2019

Pseudo-Einstein structure, eigenvalue estimate for the CR Paneitz operator and its applications to uniformization theorem

In this note, we mainly focus on the existence of pseudo-Einstein contact forms, an upper bound eigenvalue estimate for the CR Paneitz operator and its applications to the uniformization theorem for Sasakian space form in an embeddable closed strictly pseudoconvex CR 3-manifold. Firstly, the existence of pseudo-Einstein contact form is confirmed if the CR 3-manifold is Sasakian. Secondly, we derive an eigenvalue upper bound estimate for the CR Paneitz operator and obtain the CR uniformization theorem for a class of CR 3-manifolds. At the end, under the positivity assumption of the pseudohermitian curvature, we derive the existence theorem for pseudo-Einstein contact forms and uniformization theorems in a closed strictly pseudoconvex CR 3-manifold of nonnegative CR Paneitz operator with kernel consisting of the CR-pluriharmonic functions and the CR Q-curvature is CR-pluriharmonic.

math.DG↗
arXiv2018

The geometry of generalized Lamé equation, II: Existence of pre-modular forms and application

In this paper, the second in a series, we continue to study the generalized Lamé equation with the Treibich-Verdier potential \begin{equation*} y^{\prime \prime }(z)=\bigg[ \sum_{k=0}^{3}n_{k}(n_{k}+1)\wp(z+\tfrac{ ω_{k}}{2}|τ)+B\bigg] y(z),\quad n_{k}\in \mathbb{Z}_{\geq0} \end{equation*} from the monodromy aspect. We prove the existence of a pre-modular form $Z_{r,s}^{\mathbf{n}}(τ)$ of weight $\frac{1}{2}\sum n_k(n_k+1)$ such that the monodromy data $(r,s)$ is characterized by $Z_{r,s}^{\mathbf{n}}(τ)=0$. This generalizes the result in \cite{LW2}, where the Lamé case (i.e. $n_1=n_2=n_3=0$) was studied by Wang and the third author. As applications, we prove among other things that the following two mean field equations \[Δu+e^u=16πδ_{0}\quad\text{and}\quad Δu+e^u=8π\sum_{k=1}^3δ_{\frac{ω_k}{2}}\] on a flat torus $E_τ:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}τ)$ has the same number of even solutions. This result is quite surprising from the PDE point of view.

math.CA↗
arXiv2018

CR Sub-Laplacian Comparison and Liouville-type Theorem in a Complete Noncompact Sasakian Manifold

In this paper, we first obtain the sub-Laplacian comparison theorem in a complete noncompact pseudohermitian manifold of vanishing torsion (i.e. Sasakian manifold). Secondly, we derive the sub-gradient estimate for positive pseudoharmonic functions in a complete noncompact pseudohermitian manifold which satisfies the CR sub-Laplacian comparison property. It is served as the CR analog of Yau's gradient estimate. As a consequence, we have the natural CR analogue of Liouville-type theorems in a complete noncompact Sasakian manifold of nonnegative pseudohermitian Ricci curvature tensors.

math.AP↗
arXiv2017

The geometry of generalized Lamé equation, I

In this paper, we prove that the spectral curve $Γ_{\mathbf{n}}$ of the generalized Lamé equation with the Treibich-Verdier potential \begin{equation*} y^{\prime \prime }(z)=\bigg[ \sum_{k=0}^{3}n_{k}(n_{k}+1)\wp(z+\tfrac{% ω_{k}}{2}|τ)+B\bigg] y(z),\text{ \ }n_{k}\in \mathbb{Z}_{\geq0} \end{equation*} can be embedded into the symmetric space Sym$^{N}E_τ$ of the $N$-th copy of the torus $E_τ$, where $N=\sum n_{k}$. This embedding induces an addition map $σ_{\mathbf{n}}(\cdot|τ)$ from $Γ_{\mathbf{n}}$ onto $E_τ$. The main result is to prove that the degree of $σ_{% \mathbf{n}}(\cdot|τ)$ is equal to% \begin{equation*} \sum_{k=0}^{3}n_{k}(n_{k}+1)/2. \end{equation*} This is the first step toward constructing the premodular form associated with this generalized Lamé equation.

math.CA↗
arXiv2017

Non-existence of solutions for a mean field equation on flat tori at critical parameter $16π$

It is known from \cite{LW} that the solvability of the mean field equation $Δu+e^{u}=8nπδ_{0}$ with $n\in\mathbb{N}_{\geq 1}$ on a flat torus $E_τ$ essentially depends on the geometry of $E_τ$. A conjecture is the non-existence of solutions for this equation if $E_τ$ is a rectangular torus, which was proved for $n=1$ in \cite{LW}. For any $n\in \mathbb{N}_{\geq2}$, this conjecture seems challenging from the viewpoint of PDE theory. In this paper, we prove this conjecture for $n=2$ (i.e. at critical parameter $16π$).

math.AP↗
arXiv2017

Simple zero property of some holomorphic functions on the moduli space of tori

We prove that some holomorphic functions on the moduli space of tori have only simple zeros. Instead of computing the derivative with respect to the moduli parameter $τ$, we introduce a conceptual proof by applying Painlevé VI\ equation. As an application of this simple zero property, we obtain the smoothness of all the degeneracy curves of trivial critical points for some multiple Green function.

math.CV↗
arXiv2017

Unitary monodromy implies the smoothness along the real axis for some Painlevé VI equation, I

In this paper, we study the Painlevé VI equation with parameter $(\frac {9}{8},\frac{-1}{8},\frac{1}{8},\frac{3}{8})$. We prove (i) An explicit formula to count the number of poles of an algebraic solution with the monodromy group $D_{N}$, where $D_{N}$ is the dihedral group of order $2N$. (ii) There are only four solutions without poles in $\mathbb{C}\backslash \left \{ 0,1\right \} $. (iii) If the monodromy group of the associated linear ODE of a solution $λ\left( t\right) $ is unitary, then $λ( t) $ has no poles in $\mathbb{R}% \backslash \{ 0,1\} $.

math.CA↗