Chang‐Shou Lin
YOU?
Author Swipe
View article: Sovability of curvature equations with multiple singular sources on torus via Painleve VI equations
Sovability of curvature equations with multiple singular sources on torus via Painleve VI equations Open
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…
View article: On the closure of the Airault–Mckean–Moser locus for elliptic KdV potentials via Darboux transformations
On the closure of the Airault–Mckean–Moser locus for elliptic KdV potentials via Darboux transformations Open
We study the general elliptic KdV potentials, which can be expressed (up to adding a constant) as q_{\mathbf{p}}(z)\coloneq\sum_{j=1}^{n}m_{j}(m_{j}+1)\wp(z-p_{j}),\quad m_{j}\in\mathbb{N}. We give an elementary proof of the theorem that t…
View article: Critical points of the Eisenstein series $E_{4}$ and application to the spectrum of the Lamé operator
Critical points of the Eisenstein series $E_{4}$ and application to the spectrum of the Lamé operator Open
We give a complete description of the distribution of the critical points of the classical Eisenstein series E_{4}(\tau) . An application to the spectrum of the Lamé operator is also given.
View article: Co-Axial Metrics on the Sphere and Algebraic Numbers
Co-Axial Metrics on the Sphere and Algebraic Numbers Open
In this paper, we consider the following curvature equation $$\Delta u+{\rm e}^u=4\pi\biggl((\theta_0-1)\delta_0+(\theta_1-1)\delta_1 +\sum_{j=1}^{n+m}\bigl(\theta_j'-1\bigr)\delta_{t_j}\biggr)\qquad \text{in}\ \mathbb R^2,$$ $$u(x)=-2(1+\…
View article: Monodromy of generalized Lame equations with Darboux-Treibich-Verdier potentials: A universal law
Monodromy of generalized Lame equations with Darboux-Treibich-Verdier potentials: A universal law Open
The Darboux-Treibich-Verdier (DTV) potential $\sum_{k=0}^{3}n_{k}(n_{k}+1)\wp(z+\tfrac{ ω_{k}}{2};τ)$ is well-known as doubly-periodic solutions of the stationary KdV hierarchy (Treibich-Verdier, Duke Math. J. {\bf 68} (1992), 217-236). In…
View article: On the first eigenvalue of Liouville-type problems
On the first eigenvalue of Liouville-type problems Open
The aim of this note is to study the spectrum of a linearized Liouville-type problem, characterizing the case in which the first eigenvalue is zero. Interestingly enough, we obtain also point-wise information on the associated first eigenf…
View article: The tt*-Toda equations of A_n type
The tt*-Toda equations of A_n type Open
In previous articles we have studied the A_n tt*-Toda equations (topological-antitopological fusion equations of Toda type) of Cecotti and Vafa, giving details mainly for n=3. Here we give a proof of the existence and uniqueness of global …
View article: Non-degeneracy and uniqueness of solutions to general singular Toda systems on bounded domains
Non-degeneracy and uniqueness of solutions to general singular Toda systems on bounded domains Open
In this note we show non-degeneracy and uniqueness results for solutions of Toda systems associated to general simple Lie algebras with multiple singular sources on bounded domains. The argument is based on spectral properties of Cartan ma…
View article: Co-Axial Metrics on the Sphere and Algebraic Numbers
Co-Axial Metrics on the Sphere and Algebraic Numbers Open
In this paper, we consider the following curvature equation $$Δu+{\rm e}^u=4π\biggl((θ_0-1)δ_0+(θ_1-1)δ_1 +\sum_{j=1}^{n+m}\bigl(θ_j'-1\bigr)δ_{t_j}\biggr)\qquad \text{in}\ \mathbb R^2,$$ $$u(x)=-2(1+θ_\infty)\ln|x|+O(1)\qquad \text{as} \ …
View article: Monodromy of a generalized Lame equation of third order
Monodromy of a generalized Lame equation of third order Open
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 $τ$…
View article: Modular Ordinary Differential Equations on SL(2,Z) of Third Order and Applications
Modular Ordinary Differential Equations on SL(2,Z) of Third Order and Applications Open
In this paper, we study third-order modular ordinary differential equations\n(MODE for short) of the following form $y'''+Q_2(z)y'+Q_3(z)y=0$,\n$z\\in\\mathbb{H}=\\{z\\in\\mathbb{C} \\,|\\,\\operatorname{Im}z>0 \\}$, where $Q_2(z)$\nand $Q…
View article: Metrics with positive constant curvature and modular differential equations
Metrics with positive constant curvature and modular differential equations Open
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 prob…
View article: On number and evenness of solutions of the $SU(3)$ Toda system on flat tori with non-critical parameters
On number and evenness of solutions of the $SU(3)$ Toda system on flat tori with non-critical parameters Open
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:=\mat…
View article: Modular ordinary differential equations on $\mathrm{SL}(2,\mathbb{Z})$ of third order and applications
Modular ordinary differential equations on $\mathrm{SL}(2,\mathbb{Z})$ of third order and applications Open
In this paper, we study third-order modular ordinary differential equations (MODE for short) of the following form \[y'''+Q_2(z)y'+Q_3(z)y=0,\quad z\in\mathbb{H}=\{z\in\mathbb{C} \,|\,\operatorname{Im}z>0 \},\] where $Q_2(z)$ and $Q_3(z)-\…
View article: Proof of a conjecture of Dahmen and Beukers on counting integral Lamé equations with finite monodromy
Proof of a conjecture of Dahmen and Beukers on counting integral Lamé equations with finite monodromy Open
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( \fr…
View article: Quasimodular forms and modular differential equations which are not apparent at cusps: I
Quasimodular forms and modular differential equations which are not apparent at cusps: I Open
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 …
View article: The geometry of generalized Lame equation, III: One-to-one of the Riemann-Hilbert correspondence
The geometry of generalized Lame equation, III: One-to-one of the Riemann-Hilbert correspondence Open
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+\t…
View article: Blow up at infinity in the SU(3) Chern-Simons model, part I
Blow up at infinity in the SU(3) Chern-Simons model, part I Open
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…
View article: A necessary and sufficient condition for the Darboux-Treibich-Verdier potential with its spectrum contained in $\mathbb{R}$
A necessary and sufficient condition for the Darboux-Treibich-Verdier potential with its spectrum contained in $\mathbb{R}$ Open
In this paper, we study the spectrum of the complex Hill operator $L=\frac{d^2}{dx^2}+q(x;τ)$ in $L^2(\mathbb{R},\mathbb{C})$ with the Darboux-Treibich-Verdier potential \[q(x;τ):=-\sum_{k=0}^{3}n_{k}(n_{k}+1)\wp \left( x+z_0+\tfrac{ω_{k}}…
View article: Existence of bubbling solutions without mass concentration
Existence of bubbling solutions without mass concentration Open
The seminal work by Brezis and Merle has been pioneering in studying the bubbling phenomena of the mean field equation with singular sources. When the vortex points are not collapsing, the mean field equation possesses the property of the …
View article: The geometry of generalized Lamé equation, II: Existence of pre-modular forms and application
The geometry of generalized Lamé equation, II: Existence of pre-modular forms and application Open
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…
View article: Existence of bubbling solutions without mass concentration
Existence of bubbling solutions without mass concentration Open
The seminal work \cite{bm} by Brezis and Merle has been pioneering in studying the bubbling phenomena of the mean field equation with singular sources. When the vortex points are not collapsing, the mean field equation possesses the proper…
View article: Toda systems and hypergeometric equations
Toda systems and hypergeometric equations Open
This paper establishes certain existence and classification results for solutions to $\mathrm {SU}(n)$ Toda systems with three singular sources at 0, 1, and $\infty$. First, we determine the necessary conditions for such an $\mathrm {SU}(n…
View article: Sharp nonexistence results for curvature equations with four singular sources on rectangular tori
Sharp nonexistence results for curvature equations with four singular sources on rectangular tori Open
In this paper, we prove that there are no solutions for the curvature equation \[ Δu+e^{u}=8πnδ_{0}\text{ on }E_τ, \quad n\in\mathbb{N}, \] where $E_τ$ is a flat rectangular torus and $δ_{0}$ is the Dirac measure at the lattice points. Thi…
View article: The geometry of generalized Lamé equation, I
The geometry of generalized Lamé equation, I Open
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\…
View article: Degree counting for Toda system with simple singularity : one point blow up
Degree counting for Toda system with simple singularity : one point blow up Open
In this paper, we study the degree counting formula of the rank two Toda system with simple singular source when $ρ_1\in(0,4π)\cup(4π,8π)$ and $ρ_2\notin 4π\mathbb{N}.$ The key step is to derive the degree formula of the shadow system, whi…
View article: Uniqueness for bubbling solutions with collapsing singularities
Uniqueness for bubbling solutions with collapsing singularities Open
The seminal work \cite{bm} by Brezis and Merle showed that the bubbling solutions of the mean field equation have the property of mass concentration. Recently, Lin and Tarantello in \cite{lt} found that the "bubbling implies mass concentra…
View article: Critical points of the classical Eisenstein series of weight two
Critical points of the classical Eisenstein series of weight two Open
In this paper, we completely determine the critical points of the normalized Eisenstein series $E_2(τ)$ of weight $2$. Although $E_2(τ)$ is not a modular form, our result shows that $E_2(τ)$ has at most one critical point in every fundamen…