arXiv subjects

Erjuan Fu

Publications and source records attributed to Erjuan Fu.

10 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

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↗

arXiv2022

Tube classes over elementary vanishing cycles

Let $X$ be a closed Riemann surface. When $X$ is embedded into a projective space, the first rational cohomology group can be concretely obtained from the monodromy in the family of its smooth hyperplane sections by C. Schnell's tube mapping. We generalize this result to the first integral homology group by relating the tube mapping with the topological Abel--Jacobi mapping. By making use of the mapping class group action, we prove that all tube classes constructed from the elementary vanishing cycles form a cofinite subgroup of the first integral homology group of $X$.

math.AG↗

arXiv2022

A new approach towards Lefschetz $(1, 1)$-Theorem

Let $S$ be a complex projective surface. Lefschetz originally proved Lefschetz $(1, 1)$--Theorem by studying a Lefschetz pencil of hyperplane sections of $S$ and the Abel--Jacobi mapping. In this paper, we attack Lefschetz $(1, 1)$--Theorem by constructing certain two-parameter families of twice hyperplane sections of $S$ and then applying the topological Abel--Jacobi mapping. Our geometric constructions would give an inductive approach and some insight for higher dimensional cases. We prove a strong tube theorem which generalizes Schnell's tube theorem to integral homology groups for complex projective curves and then obtain a Jacobi-type inversion theorem. In the end, we give a geometric description for the deformation space of an elementary vanishing cycle over a generic net.

math.AG↗

arXiv2021

Spectrum of the Lam\'{e} operator along $\mathrm{Re}\tau={1}/{2}:$ The genus $3$ case

In this paper, we study the spectrum $\sigma(L)$ of the Lam\'{e} operator \begin{equation*}L=\frac{d^2}{dx^2}-12\wp(x+z_0;\tau)\quad \text{in}\;\;L^2(\mathbb{R}, \mathbb{C}), \end{equation*} where $\wp(z;\tau)$ is the Weierstrass elliptic function with periods $1$ and $\tau$, and $z_0\in\mathbb{C}$ is chosen such that $L$ has no singularities on $\mathbb{R}$. We prove that a point $\lambda\in \sigma(L)$ is an intersection point of different spectral arcs but not a zero of the spectral polynomial if and only if $\lambda$ is a zero of the following cubic polynomial: \begin{equation*} \frac{4}{15} \lambda^3+\frac{8}{5}\eta_1 \lambda^2-3g_2 \lambda+9g_3-6\eta_1 g_2=0. \end{equation*} We also study the deformation of the spectrum as $\tau=\frac{1}{2}+ib$ with $b>0$ varying. We discover $7$ different types of graphs for the spectrum as $b$ varies around the double zeros of the spectral polynomial.

math.CA↗

arXiv2020

A necessary and sufficient condition for the Darboux-Treibich-Verdier potential with its spectrum contained in $\mathbb{R}$

In this paper, we study the spectrum of the complex Hill operator $L=\frac{d^2}{dx^2}+q(x;\tau)$ in $L^2(\mathbb{R},\mathbb{C})$ with the Darboux-Treibich-Verdier potential \[q(x;\tau):=-\sum_{k=0}^{3}n_{k}(n_{k}+1)\wp \left( x+z_0+\tfrac{\omega_{k}}{2};\tau \right),\] where $n_k\in\mathbb{Z}_{\geq 0}$ with $\max n_k\geq 1$ and $z_0\in\mathbb{C}$ is chosen such that $q(x;\tau)$ has no singularities on $\mathbb{R}$. For any fixed $\tau\in i\mathbb{R}_{>0}$, we give a necessary and sufficient condition on $(n_0,n_1,n_2,n_3)$ to guarantee that the spectrum $\sigma(L)$ is \[\sigma(L)=(-\infty, E_{2g}]\cup[E_{2g-1}, E_{2g-2}]\cup \cdots \cup[E_{1}, E_{0}],\quad E_j\in \mathbb{R},\] and hence generalizes Ince's remarkable result in 1940 for the Lam\'{e} potential to the Darboux-Treibich-Verdier potential. We also determine the number of (anti)periodic eigenvalues in each bounded interval $(E_{2j-1}$, $E_{2j-2})$, which generalizes the recent result in \cite{HHV} where the Lam\'{e} case $n_1=n_2=n_3=0$ was studied.

math.CA↗