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Yiqian Shi

Publications and source records attributed to Yiqian Shi.

14 recordsLinked to original sources
arXiv2026

Zariski-Dense Monodromy of Singular Hyperbolic Metrics on Non-Hyperbolic Riemann Surfaces

We prove that the monodromy group of every singular hyperbolic metric on a non-hyperbolic Riemann surface, in the sense of potential theory, is Zariski dense in ${\rm PSL}(2,\mathbb{R})$, confirming a conjecture of the authors. The main new step is to show that a singular hyperbolic metric on an arbitrary parabolic Riemann surface cannot have monodromy contained in a conjugate of the real affine subgroup of ${\rm PSL}(2,\mathbb{R})$. The same argument also gives a direct proof in the compact case. Combined with the nonexistence results for the remaining proper subgroup types, this proves the conjecture.

math.DG↗
arXiv2025

The Hurwitz existence problem and the prime-degree conjecture: A computational perspective

We investigate the Hurwitz existence problem from a computational viewpoint. Leveraging the symmetric-group algorithm by Zheng and building upon implementations originally developed by Baroni, we achieve a complete and non-redundant enumeration of all non-realizable partition triples for positive integers up to $31$. These results are further categorized into four types according to their underlying mathematical structure; it is observed that nearly nine-tenths of them can be explained by known theoretical results. As an application, we verify the prime-degree conjecture for all primes less than $32$. In light of the exponential memory growth inherent in existing computational approaches -- which limits their feasibility at higher degrees -- we propose a novel software architecture designed to stabilize memory usage, thereby facilitating further detection of exceptional cases in the Hurwitz existence problem. The complete dataset of non-realizable partition triples, along with our implementation, will been made public on GitHub.

math.GR↗
arXiv2025

Solution to SU(n+1) Toda system generated by spherical metrics

Using the correspondence between solutions to the SU(n+1) Toda system on a Riemann surface and totally unramified unitary curves, we show that a spherical metric $ω$ generates a family of solutions, including $(i(n+1-i)ω)_{i=1}^n$. Moreover, we characterize this family in terms of the monodromy group of the spherical metric. As a consequence, we obtain a new solution class to the SU(n+1) Toda system with cone singularities on compact Riemann surfaces, complementing the existence results of Lin-Yang-Zhong (JDG, 114(2):337-391, 2020).

math-ph↗
arXiv2024

Classifying solutions of ${\rm SU}(n+1)$ Toda system around a singular source

Consider a positive integer $n$ and $γ_1>-1,\cdots,γ_n>-1$. Let $D=\{z\in {\Bbb C}:|z|<1\}$, and let $(a_{ij})_{n\times n}$ denote the Cartan matrix of $\frak{su}(n+1)$. Utilizing the ordinary differential equation of $(n+1)$th order around a singular source of ${\rm SU}(n+1)$ Toda system, as discovered by Lin-Wei-Ye ({\it Invent Math}, {\bf 190}(1):169-207, 2012), we precisely characterize a solution $(u_1,\cdots, u_n)$ to the ${\rm SU}(n+1)$ Toda system \begin{equation*} \begin{cases} \frac{\partial^2 u_i}{\partial z\partial \bar z}+\sum_{j=1}^n a_{ij} e^{u_j}&=πγ_iδ_0\,\,{\rm on}\,\, D\\ \frac{\sqrt{-1}}{2}\,\int_{D\backslash \{0\}} e^{u_{i} }{\rm d}z\wedge {\rm d}\bar z &< \infty \end{cases} \quad \text{for all}\quad i=1,\cdots, n \end{equation*} using $(n+1)$ holomorphic functions that satisfy the normalized condition. Additionally, we demonstrate that for each $1\leq i\leq n$, $0$ represents the cone singularity with angle $2π(1+γ_i)$ for the metric $e^{u_i}|{\rm d}z|^2$ on $D\backslash\{0\}$, which can be locally characterized by $(n-1)$ non-vanishing holomorphic functions at $0$.

math.AP↗
arXiv2024

Solutions to ${\rm SU}(n+1)$ Toda system with cone singularities via toric curves on compact Riemann surfaces

On a compact Riemann surface (X) with finite punctures (P_1, \ldots, P_k), we define toric curves as multi-valued, totally unramified holomorphic maps to (\mathbb{P}^n) with monodromy in a maximal torus of ({\rm PSU}(n+1)). \textit{Toric solutions} for the ({\rm SU}(n+1)) system on $X\setminus\{P_1,\ldots, P_k\}$ are recognized by their associated {\it toric} curves in (\mathbb{P}^n). We introduce a character n-ensemble as an (n)-tuple of meromorphic one-forms with simple poles and purely imaginary periods, generating toric curves on (X) minus finitely many points. We establish on $X$ a correspondence between character $n$-ensembles and toric solutions to the ({\rm SU}(n+1)) system with finitely many cone singularities. Our approach not only broadens seminal solutions for up to two cone singularities on the Riemann sphere, as classified by Jost-Wang (Int. Math. Res. Not., (6):277-290, 2002) and Lin-Wei-Ye (Invent. Math., 190(1):169-207, 2012), but also advances beyond the limits of Lin-Yang-Zhong's existence theorems (J. Differential Geom., 114(2):337-391, 2020) by introducing a new solution class.

math.DG↗
arXiv2023

The space of germs of extremal K\" ahler metrics in one dimension comprises three distinct ${\Bbb R}^3$ components

In the 1980s, Eugenio Calabi introduced the concept of {\it extremal K\" ahler metrics} as critical points of the $L^2$-norm functional of scalar curvature in the space of K\" ahler metrics belonging to a fixed Kähler class of a compact complex manifold $X$. Calabi demonstrated that extremal K\" ahler metrics always degenerate into Einstein metrics on compact Riemann surfaces. We define a Kähler metric $g$ on a domain of ${\Bbb C}^n$ as a {\it local extremal Kähler metric} of dimension $n$ if it satisfies the Euler-Lagrange equation of this functional, i.e. holomorphic is the $(1,0)$-part of the gradient vector field of the scalar curvature of $g$, in the domain. Our main result establishes that the space of all germs of local extremal, non-Einstein Kähler metrics of dimension one comprises three components, each diffeomorphic to ${\Bbb R}^3$.

math.DG↗
arXiv2022

Solutions of the ${\rm SU}(n+1)$ Toda system from meromorphic functions

We consider the ${\rm SU}(n+1)$ Toda system on a simply connected domain $Ω$ in ${\Bbb C}$, the $n=1$ case of which coincides with the Liouville equation $Δu+8e^u=0$. A classical result by Liouville says that a solution of this equation on $Ω$ can be represented by some non-degenerate meromorphic function on $Ω$. We construct a family of solutions parameterized by ${\rm PSL}(n+1,\,{\Bbb C})/{\rm PSU}(n+1)$ for the ${\rm SU}(n+1)$ Toda system from such a meromorphic function on $Ω$, which generalizes the result of Liouville. As an application, we find a new class of solvable ${\rm SU}(n+1)$ Toda systems with singular sources via cone spherical metrics on compact Riemann surfaces.

math-ph↗
arXiv2020

Singular hyperbolic metrics and negative subharmonic functions

We propose a conjecture that the monodromy group of a singular hyperbolic metric on a non-hyperbolic Riemann surface is {\it Zariski dense} in ${\rm PSL}(2,\,{\Bbb R})$. By using meromorphic differentials and affine connections, we obtain an evidence of the conjecture that the monodromy group of the singular hyperbolic metric can not be contained in four classes of one-dimensional Lie subgroups of ${\rm PSL}(2,\,{\Bbb R})$. Moreover, we confirm the conjecture if the Riemann surface is either one of the once punctured Riemann sphere, the twice punctured Riemann sphere, a once punctured torus and a compact Riemann surface.

math.DG↗
arXiv2018

Isolated singularities of conformal hyperbolic metrics

J. Nitsche proved that an isolated singularity of a conformal hyperbolic metric is either a conical singularity or a cusp one. We prove by developing map that there exists a complex coordinate $z$ centered at the singularity where the metric has the expression of either $\displaystyle{\frac{4α^2\vert z \vert^{2α-2}}{(1-\vert z \vert ^{2α})^2}\vert \mathrm{d} z \vert^2}$ with $α>0$ or $\displaystyle{\vert z \vert ^{-2}\big(\ln|z|\big)^{-2}|dz|^{2}}$.

math.DG↗
arXiv2013

Gradient estimate of a Neumann eigenfunction on a compact manifold with boundary

Let $e_ł(x)$ be a Neumann eigenfunction with respect to the positive Laplacian $Δ$ on a compact Riemannian manifold $M$ with boundary such that $Δ\, e_ł=ł^2 e_ł$ in the interior of $M$ and the normal derivative of $e_ł$ vanishes on the boundary of $M$. Let $χ_λ$ be the unit band spectral projection operator associated with the Neumann Laplacian and $f$ a square integrable function on $M$. We show the following gradient estimate for $χ_λ\,f$ as $λ\geq 1$: $\|\nabla\ χ_ł f\|_\infty\leq Cł\|χ_ł\f\|_\infty+ł^{-1}\|Δ χ_ł f\|_\infty$, where $C$ is a positive constant depending only on $M$. As a corollary, we obtain the gradient estimate of $e_ł$: for every $ł\geq 1$, there holds $\|\nabla e_ł\|_\infty\leq C\,ł\, \|e_ł\|_\infty$.

math.SP↗
arXiv2010

Gradient estimate of a Dirichlet eigenfunction on a compact manifold with boundary

Let $e_ł(x)$ be an eigenfunction with respect to the Dirichlet Laplacian $Δ_N$ on a compact Riemannian manifold $N$ with boundary: $Δ_N e_ł=ł^2 e_ł$ in the interior of $N$ and $e_ł=0$ on the boundary of $N$. We show the following gradient estimate of $e_ł$: for every $ł\geq 1$, there holds $ł\|e_ł\|_\infty/C\leq \|\nabla e_ł\|_\infty\leq Cł\|e_ł\|_\infty$, where $C$ is a positive constant depending only on $N$. In the proof, we use a basic geometrical property of nodal sets of eigenfunctions and elliptic apriori estimates.

math.AP↗
arXiv2009

Gradient estimate of an eigenfunction on a compact Riemannian manifold without boundary

Let $e_ł(x)$ be an eigenfunction with respect to the Laplace-Beltrami operator $Δ_M$ on a compact Riemannian manifold $M$ without boundary: $Δ_M e_ł=ł^2 e_ł$. We show the following gradient estimate of $e_ł$: for every $ł\geq 1$, there holds $ł\|e_ł\|_\infty/C\leq \|\nabla e_ł\|_\infty\leq Cł\|e_ł\|_\infty$, where $C$ is a positive constant depending only on $M$.

math.SP↗
arXiv2009

The Riemannian manifolds with boundary and large symmetry

Sixty years ago, S. B. Myers and N. E. Steenrod ({\it Ann. of Math.} {\bf 40} (1939), 400-416) showed that the isometry group of a Riemannian manifold without boundary has a structure of Lie group. Recently A. V. Bagaev and N. I. Zhukova ({\it Siberian Math. J.} {\bf 48} (2007), 579-592) proved the same result for a Riemannian orbifold. In this paper, we firstly show that the isometry group of a Riemannian manifold $M$ with boundary has dimension at most ${1/2} \dim M (\dim M-1)$. Then we completely classify such Riemannian manifolds with boundary that their isometry groups attain the preceding maximal dimension.

math.DG↗