arXiv subjects

Chang-Shou Lin

Publications and source records attributed to Chang-Shou Lin.

At least 19 recordsLinked to original sources

arXiv2026

Classification of spherical metrics on tori with four singularities, I: half periods

Classifying the spherical metrics on a torus $E_\tau$ with $4\pi$ conic angle at each half period point\, ${\omega_k}/{2}, k=0,1,2,3$\, is equivalent to classify solutions of the following curvature equation \begin{align}\label{eq0731093154} \Delta u+e^u=4\pi\sum_{k=0}^3\delta_{\frac{\omega_k}{2}}\text{\ on\ }E_\tau \end{align} where $\tau\in \mathbb{H}:=\{z\in \mathbb{C}\mid \mathrm{Im} \, z>0\}$ and $\delta_p$ is the Dirac measure at $p\in E_\tau$. By constructing a multiple Green function $$G_2(z_1, z_2;\tau):=G(z_1-z_2;\tau)-\frac{1}{2}\sum_{j=0}^3\left(G(z_1-\frac{\omega_j}{2};\tau)+G(z_2-\frac{\omega_j}{2};\tau)\right), $$ in terms of the Green function $G(z;\tau)$ on $E_\tau$, we classify the solutions of (\ref{eq0731093154}) into two types: \emph{special} and \emph{non-special}. Furthermore, we obtain the following conclusion about the solutions of (\ref{eq0731093154}): \begin{enumerate} \item any special solution is an even function and the set of special solutions is isomorphic to $SL(2,\mathbb{C})/SU(2)$ for all $\tau\in \mathbb{H}$. \item a non-special solution exists if and only if $\tau\in \mathcal{E}$. Moreover, if $\tau\in \mathcal{E}$, then there are six one-parameter families of nonspecial solutions. \end{enumerate} where $$\mathcal{E}:=\left\{\tau\in \mathbb{H}\mid G(z;\tau) \,\,\text{has exactly 5 critical points.}\right\}.$$ The set $\mathcal{E}$ is completely determined in \cite{CLW2018, Lin}, which is a union of countable many open triangular domains. As a byproduct, we completely determine and classify the critical points of $G_2$ and then obtain the degeneracy criterion of critical points for $G_2$, which may be of independent interest.

math.AP↗

arXiv2026

On monodromy and spectral geometry of generalized Lam\'e equations with four singularities, I: half periods

We consider the unitary monodromy problem of the following generalized Lam\'e equations with apparent parameters \begin{equation*} y^{\prime\prime}(z)=\left(\frac{3}{4}\sum_{k=0}^3\wp(z-\frac{\omega_k}{2};\tau)+\sum_{k=0}^3T_k\zeta(z-\frac{\omega_k}{2};\tau)+B\right)y(z), \end{equation*} where $T_0,\cdots, T_3, B$ are apparent parameters. We first decompose the space of apparent parameters, which turns out to be an algebraic set, into three irreducible components. These three components intersect at $(T_0,\cdots, T_3, B)=(0,\cdots, 0)$, which plays an important role in determining whether the monodromy matrices is unitary or not. Following the approach in KdV theory, we define the spectral polynomial which is a degree 4 polynomial of the apparent parameter. We then obtain that the monodromy is not completely reducible if and only if the apparent parameter is a zero of the spectral polynomial. By introducing a branched double cover of the apparent space, which parametrizes all one-dimensional common eigenspaces, we determine the monodromy data for all apparent parameters. By noticing that the equation under the covering map is exactly the spectral polynomial, we obtain that the generalized Lam\'e curve is isomorphic to the spectral curve. Finally, with the help of the spectral curve defined by the spectral polynomial, we characterize the conditional stability sets in two directions by making use of the local analytic coordinates of the monodromy data and then prove that the monodromy matrices are unitary if and only if $(T_0,\cdots, T_3, B)=(0,\cdots, 0)$ when the period $\tau\in i\mathbb{R}_{>0}$.

math.CA↗

arXiv2026

Geometric analysis on rhombus torus: Green function with two singularities

Let $G(z)$ be the Green function on the flat torus $E_{\tau}=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau)$ with the singularity at $0$. Lin and Wang (Ann. Math. 2010) proved that $G(z)$ has at most one pair of nontrivial critical points. This is the third of a series of papers to study the sum of two Green functions which can be reduced to $G_p(z):=\frac12(G(z+p)+G(z-p))$. We study how the geometry of the torus and the location of singularities $\pm p$ affect the structure of critical points of $G_p(z)$. In Part I \cite{CFL}, we proved that $G_p(z)$ has at most three pairs of nontrivial critical points for all tori. In Part II \cite{CFL-II} (Proc. Lond. Math. Soc. 2026), we studied the important case that $E_{\tau}$ is a rectangular torus. In this paper, first we prove that if $G_p(z)$ has three pairs of nontrivial critical points, then critical points are all non-degenerate. Secondly, we study the other important but more challenging case that $E_{\tau}$ is a rhombus torus, by developing different approaches from \cite{CFL, CFL-II}. As applications, we show that the curvature equation $\Delta u+e^{u}=4\pi(\delta_p+\delta_{-p})$ on $E_{\tau}$ has exactly either $0$, $1$ or $2$ even axisymmetric solutions and each number really occurs.

math.AP↗

arXiv2025

Green functions, Hitchin's formula and curvature equations on tori II: Rectangular torus

Let $G(z)$ be the Green function on the flat torus $E_{\tau}=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau)$ with the singularity at $0$. Lin and Wang (Ann. Math. 2010) proved that $G(z)$ has either $3$ or $5$ critical points (depending on the choice of $\tau$). Here we study the sum of two Green functions which can be reduced to $G_p(z):=\frac12(G(z+p)+G(z-p))$. In Part I \cite{CFL}, we proved that for any $p$ satisfying $p\neq -p$ in $E_{\tau}$, the number of critical points of $G_p(z)$ belongs to $\{4,6,8,10\}$ (depending on the choice of $(\tau, p)$) and each number really occurs. In the Part II of this series, we study the important case $\tau=ib$ with $b>0$, i.e. $E_{\tau}$ is a rectangular torus. By developing a completely different approach from Part I, we show the existence of $8$ real values $d_1<d_2<\cdots<d_7<d_8$ such that if $$\wp(p)\in (-\infty, d_1]\cup [d_2, d_3]\cup [d_4, d_5]\cup [d_6, d_7]\cup [d_8,+\infty),$$ then $G_p(z)$ has no nontrivial critical points; if $$\wp(p)\in (d_1, d_2)\cup (d_3, d_4)\cup (d_5, d_6)\cup (d_7, d_8),$$ then $G_p(z)$ has a unique pair of nontrivial critical points that are always non-degenerate saddle points. This allows us to study the possible distribution of the numbers of critical points of $G_p(z)$ for generic $p$. Applications to the Painlev\'{e} VI equation and the curvature equation are also given.

math.AP↗

arXiv2025

Green functions, Hitchin's formula and curvature equations on tori

Let $G(z)=G(z;\tau)$ be the Green function on the flat torus $E_{\tau}=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau)$ with the singularity at $0$. Lin and Wang (Ann. Math. 2010) proved that $G(z)$ has either $3$ or $5$ critical points (depending on the choice of $\tau$). Later, Bergweiler and Eremenko (Proc. Amer. Math. Soc. 2016) gave a new proof of this remarkable result by using anti-holomorphic dynamics. In this paper, firstly, we prove that once $G(z)$ has $5$ critical points, then these $5$ critical points are all non-degenerate. Secondly, we study the sum of two Green functions which can be reduced to $G_p(z):=\frac12(G(z+p)+G(z-p))$. We prove that for any $p$ satisfying $p\neq -p$ in $E_{\tau}$, the number of critical points of $G_p(z)$ belongs to $\{4,6,8,10\}$ (depending on the choice of $(\tau, p)$) and each number really occurs. We apply Hitchin's formula (J. Differ. Geom. 1995) in a surprising way to prove the generic non-degeneracy of critical points. This allows us to study the distribution of the numbers of critical points of $G_p(z)$ as $p$ varies. Applications to the curvature equation $\Delta u+e^{u}=4\pi(\delta_{p}+\delta_{-p})$ on $E_{\tau}$ are also given, and how the geometry of the torus affects the solution structure is studied.

math.AP↗

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 \[\Delta u+e^{u}=8\pi \sum_{k=0}^{3}n_{k}\delta_{\frac{\omega_{k}}{2}}% +4\pi \left( \delta_{p}+\delta_{-p}\right) \quad \text{ on }\;E_{\tau}:=\mathbb{C}/(\mathbb Z+\mathbb{Z}\tau),\] where $n_k\in\mathbb N$ and $\delta_a$ 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 Painlev\'{e} 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 Painlev\'{e} VI equations.

math.AP↗

arXiv2025

Generic non-degeneracy of critical points of multiple Green functions on torus and applications to curvature equations

Let $E_{\tau}:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau)$ with $\operatorname{Im}\tau>0$ be a flat torus and $G(z;\tau)$ be the Green function on $E_{\tau}$ with the singularity at $0$. Consider the multiple Green function $G_{n}$ on $(E_{\tau})^{n}$: \[ G_{n}(z_{1},\cdots,z_{n};\tau):=\sum_{i 0\}$ such that $G_n(\cdot;\tau)$ has degenerate critical points for any $\tau$ on the union of these curves. In this paper, we prove that there is a measure zero subset $\mathcal{O}_n\subset \mathbb H$ (containing these curves) such that for any $\tau\in \mathbb H\setminus\mathcal{O}_n$, all critical points of $G_n(\cdot;\tau)$ are non-degenerate. Applications to counting the exact number of solutions of the curvature equation $\Delta u+e^{u}=\rho \delta_{0}$ on $E_{\tau}$ will be given.

math.AP↗

arXiv2024

Monodromy of generalized Lame equations with Darboux-Treibich-Verdier potentials: A universal law

The Darboux-Treibich-Verdier (DTV) potential $\sum_{k=0}^{3}n_{k}(n_{k}+1)\wp(z+\tfrac{ \omega_{k}}{2};\tau)$ is well-known as doubly-periodic solutions of the stationary KdV hierarchy (Treibich-Verdier, Duke Math. J. {\bf 68} (1992), 217-236). In this paper, we study the generalized Lam\'{e} equation with the DTV potential \begin{equation*} y^{\prime \prime }(z)=\bigg[ \sum_{k=0}^{3}n_{k}(n_{k}+1)\wp(z+\tfrac{ \omega_{k}}{2};\tau)+B\bigg] y(z),\quad n_{k}\in \mathbb{N} \end{equation*} from the monodromy aspect. We prove that the map from $(\tau, B)$ to the monodromy data $(r,s)$ satisfies a surprising universal law $d\tau\wedge dB\equiv8\pi^2 dr\wedge ds.$ Our proof applies Panlev\'{e} VI equation and modular forms. We also give applications to the algebraic multiplicity of (anti)periodic eigenvalues for the associated Hill operator.

math.CA↗

arXiv2023

The tt*-Toda equations of A_n type

In previous articles we have studied the A_n tt*-Toda equations (topological-antitopological fusion equations of Toda type) of Cecotti and Vafa, giving details mainly for n=3. Here we give a proof of the existence and uniqueness of global solutions for any n, and a new treatment of their asymptotic data, monodromy data, and Stokes data.

math.DG↗

arXiv2023

Monodromy of a generalized Lame equation of third order

We study the monodromy of the following third order linear differential equation \[y'''(z)-(α\wp(z;τ)+B)y'(z)+β\wp'(z;τ)y(z)=0, \] where $B\in\mathbb{C}$ is a parameter, $\wp(z;τ)$ is the Weierstrass $\wp$-function with periods $1$ and $τ$, and $α,β$ are constants such that the local exponents at the singularity $0$ are three distinct integers, which can always be written as $-n-l, 1-l, n+2l+2$ after a dual transformation, where $n,l\in\mathbb{N}$. This ODE can be seen as the third order version of the well-known Lamé equation $y''(z)-(m(m+1)\wp(z;τ)+B)y(z)=0$. We say that the monodromy is unitary if the monodromy group is conjugate to a subgroup of the unitary group. We show that \begin{itemize} \item[(i)] if $n, l$ are both odd, then the monodromy can not be unitary; \item[(ii)] if $n$ is odd and $l$ is even, then there exist finite values of $B$ such that the monodromy is the Klein four-group and hence unitary; \item[(iii)] if $n$ is even, then whether there exists $B$ such that the monodromy is unitary depends on the choice of the period $τ$. \end{itemize} The methods of studying the second order Lamé equation can not work here, and we need to develop different approaches to treat these different cases separately. These results have interesting applications to the integrable $SU(3)$ Toda system in another work (Chen-Lin, J. Differ. Geom. to appear).

math.CA↗

arXiv2023

On the first eigenvalue of Liouville-type problems

The aim of this note is to study the spectrum of a linearized Liouville-type problem, characterizing the case in which the first eigenvalue is zero. Interestingly enough, we obtain also point-wise information on the associated first eigenfunction. To this end, we refine the Alexandrov-Bol inequality suitable for our problem and characterize its equality case.

math.AP↗

arXiv2023

Non-degeneracy and uniqueness of solutions to general singular Toda systems on bounded domains

In this note we show non-degeneracy and uniqueness results for solutions of Toda systems associated to general simple Lie algebras with multiple singular sources on bounded domains. The argument is based on spectral properties of Cartan matrices and eigenvalue analysis of linearized Liouville-type problems. This seems to be the first result for this class of problems and it covers all the Lie algebras of any rank.

math.AP↗

arXiv2022

Co-Axial Metrics on the Sphere and Algebraic Numbers

In this paper, we consider the following curvature equation $$\Delta u+{\rm e}^u=4\pi\biggl((\theta_0-1)\delta_0+(\theta_1-1)\delta_1 +\sum_{j=1}^{n+m}\bigl(\theta_j'-1\bigr)\delta_{t_j}\biggr)\qquad \text{in}\ \mathbb R^2,$$ $$u(x)=-2(1+\theta_\infty)\ln|x|+O(1)\qquad \text{as} \ |x|\to\infty,$$ where $\theta_0$, $\theta_1$, $\theta_\infty$, and $\theta_{j}'$ are positive non-integers for $1\le j\le n$, while $\theta_{j}'\in\mathbb{N}_{\geq 2}$ are integers for $n+1\le j\le n+m$. Geometrically, a solution $u$ gives rise to a conical metric ${\rm d}s^2=\frac12 {\rm e}^u|{\rm d}x|^2$ of curvature $1$ on the sphere, with conical singularities at $0$, $1$, $\infty$, and $t_j$, $1\le j\le n+m$, with angles $2\pi\theta_0$, $2\pi\theta_1$, $2\pi\theta_\infty$, and $2\pi\theta_{j}'$ at $0$, $1$, $\infty$, and $t_j$, respectively. The metric ${\rm d}s^2$ or the solution $u$ is called co-axial, which was introduced by Mondello and Panov, if there is a developing map $h(x)$ of $u$ such that the projective monodromy group is contained in the unit circle. The sufficient and necessary conditions in terms of angles for the existence of such metrics were obtained by Mondello-Panov (2016) and Eremenko (2020). In this paper, we fix the angles and study the locations of the singularities $t_1,\dots,t_{n+m}$. Let $A\subset\mathbb{C}^{n+m}$ be the set of those $(t_1,\dots,t_{n+m})$'s such that a co-axial metric exists, among other things we prove that (i) If $m=1$, i.e., there is only one integer $\theta_{n+1}'$ among $\theta_j'$, then $A$ is a finite set. Moreover, for the case $n=0$, we obtain a sharp bound of the cardinality of the set $A$. We apply a result due to Eremenko, Gabrielov, and Tarasov (2016) and the monodromy of hypergeometric equations to obtain such a bound. (ii) If $m\ge 2$, then $A$ is an algebraic set of dimension $\leq m-1$.

math.CA↗

arXiv2022

Modular Ordinary Differential Equations on ${\rm SL}(2,\mathbb{Z})$ of Third Order and Applications

In this paper, we study third-order modular ordinary differential equations (MODE for short) of the following form $y'''+Q_2(z)y'+Q_3(z)y=0$, $z\in\mathbb{H}=\{z\in\mathbb{C} \,|\,\operatorname{Im}z>0 \}$, where $Q_2(z)$ and $Q_3(z)-\frac12 Q_2'(z)$ are meromorphic modular forms on ${\rm SL}(2,\mathbb{Z})$ of weight $4$ and $6$, respectively. We show that any quasimodular form of depth $2$ on ${\rm SL}(2,\mathbb{Z})$ leads to such a MODE. Conversely, we introduce the so-called Bol representation $\hatρ\colon {\rm SL}(2,\mathbb{Z})\to{\rm SL}(3,\mathbb{C})$ for this MODE and give the necessary and sufficient condition for the irreducibility (resp. reducibility) of the representation. We show that the irreducibility yields the quasimodularity of some solution of this MODE, while the reducibility yields the modularity of all solutions and leads to solutions of certain ${\rm SU}(3)$ Toda systems. Note that the ${\rm SU}(N+1)$ Toda systems are the classical Plücker infinitesimal formulas for holomorphic maps from a Riemann surface to $\mathbb{CP}^N$.

math.NT↗

arXiv2021

Metrics with positive constant curvature and modular differential equations

In this paper, we consider the problem when a differential equation y"(z)=Q(z)y(z) is Fuchsian on H* and apparent on H, where Q(z) is a meromorphic modular form of weight 4 on SL(2,Z) and H denotes the complex upper half-plane. Such a problem is closely related to the problem of the existence of a conformal metric with curvature 1/2 on H.

math.CA↗

arXiv2021

On number and evenness of solutions of the $SU(3)$ Toda system on flat tori with non-critical parameters

We study the $SU(3)$ Toda system with singular sources \[ \begin{cases} Δu+2e^{u}-e^v=4π\sum_{k=0}^m n_{1,k}δ_{p_k}\quad\text{ on }\; E_τ,\\ Δv+2e^{v}-e^u=4π\sum_{k=0}^m n_{2,k}δ_{p_k}\quad\text{ on }\; E_τ, \end{cases} \] where $E_τ:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}τ)$ with $\operatorname{Im}τ>0$ is a flat torus, $δ_{p_k}$ is the Dirac measure at $p_k$, and $n_{i,k}\in\mathbb{Z}_{\geq 0}$ satisfy $\sum_{k}n_{1,k}\not\equiv \sum_k n_{2,k} \mod 3$. This is known as the non-critical case and it follows from a general existence result of \cite{BJMR} that solutions always exist. In this paper we prove that (i) The system has at most \[\frac{1}{3\times 2^{m+1}}\prod_{k=0}^m(n_{1,k}+1)(n_{2,k}+1)(n_{1,k}+n_{2,k}+2)\in\mathbb{N}\] solutions. We have several examples to indicate that this upper bound should be sharp. Our proof presents a nice combination of the apriori estimates from analysis and the classical Bézout theorem from algebraic geometry. (ii) For $m=0$ and $p_0=0$, the system has even solutions if and only if at least one of $\{n_{1,0}, n_{2,0}\}$ is even. Furthermore, if $n_{1,0}$ is odd, $n_{2,0}$ is even and $n_{1,0}<n_{2,0}$, then except for finitely many $τ$'s modulo $SL(2,\mathbb{Z})$ action, the system has exactly $\frac{n_{1,0}+1}{2}$ even solutions. Differently from \cite{BJMR}, our proofs are based on the integrability of the Toda system, and also imply a general non-existence result for even solutions of the Toda system with four singular sources.

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↗

arXiv2021

Quasimodular forms and modular differential equations which are not apparent at cusps: I

In this paper, we explore a two-way connection between quasimodular forms of depth $1$ and a class of second-order modular differential equations with regular singularities on the upper half-plane and the cusps. Here we consider the cases $Γ=Γ_0^+(N)$ generated by $Γ_0(N)$ and the Atkin-Lehner involutions for $N=1,2,3$ ($Γ_0^+(1)=\mathrm{SL}(2,\mathbb Z)$). Firstly, we note that a quasimodular form of depth $1$, after divided by some modular form with the same weight, is a solution of a modular differential equation. Our main results are the converse of the above statement for the groups $Γ_0^+(N)$, $N=1,2,3$.

math.NT↗