Yi-Hu Yang
This note is to concern a generalization to the case of twisted coefficients of the classical theory of Abelian differentials on a compact Riemann surface. We apply the Dirichlet's principle to a modified energy functional to show the existence of differentials with twisted coefficients of the second and third kinds under a suitable assumption on residues.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
The index of a meromorphic function $g$ on a compact Riemann surface is an invariant of $g$, which is defined as the number of negative eigenvalues of the differential operator $L:=-Δ-|dG|^2$, where $Δ$ is the Laplacian with respect to a conformal metric $ds^2$ on the Riemann surface, $G \colon M \to S^2$ is the holomorphic map corresponding to $g$. We consider the meromorphic function $w$ on the Riemann surface $M_a= \{(z,w) \in\widehat{\mathbb{C}}^2 \mid w^2=z(z-a)(z+\frac{1}{a})\}(a \geqslant 1 )$ homeomorphic to a torus, and we determine the index of $tw$ for all $a$ in the range $1 \leqsl…
In this paper, we analyze the theory of meromorphic $(1,0)$-forms $ω\in\mathcal{M}Ω^{(1,0)}(\mathbb{CP}^1).$ Hence, we show that on a compact Riemann surface of genus $g=0,$ isomorphic to $\mathbb{CP}^1,$ every non-constant meromorphic function $f:X\to\mathbb{CP}^1$ has as many zeros as poles, where each is counted according to multiplicities. Such an analysis gives rise to the following result. Invoking the Riemann-Roch theorem for a compact Riemann $X$ with canonical divisor $K,$ it follows that $deg(f)=0$ for any principal divisor $(f):=D$ on $X.$ More precisely, $\ell(D)-\ell(K-D)=deg(D)+1…
The Mittag-Leffler theorem for proper minimal surfaces and directed meromorphic curves
本文建立了Mittag-Leffler型定理，用于Oka锥引导的亚纯曲线和正常共形极小浸入，包含逼近与插值，并应用于证明有限总曲率极小末端的嵌入性及刻画相关开黎曼曲面。
Let $\mathsf m$ be any conical (or smooth) metric of finite volume on the Riemann sphere $\Bbb CP^1$. On a compact Riemann surface $X$ of genus $g$ consider a meromorphic funciton $f: X\to {\Bbb C}P^1$ such that all poles and critical points of $f$ are simple and no critical value of $f$ coincides with a conical singularity of $\mathsf m$ or $\{\infty\}$. The pullback $f^*\mathsf m$ of $\mathsf m$ under $f$ has conical singularities of angles $4π$ at the critical points of $f$ and other conical singularities that are the preimages of those of $\mathsf m$. We study the $ζ$-regularized determina…
We study the monodromy of meromorphic cyclic $\mathrm{SL}(n,\mathbb{C})$-opers on the Riemann sphere with a single pole. We prove that the monodromy map, sending such an oper to its Stokes data, is an immersion in the case where the order of the pole is a multiple of $n$. To do this, we develop a method based on the work of M. Jimbo, T. Miwa, and K. Ueno from the theory of isomonodromic deformations. Specifically, we introduce a system of equations that is equivalent to the isomonodromy equations of Jimbo-Miwa-Ueno, but which is adapted to the decomposition of the Lie algebra $\mathfrak{sl}(n,…
Meromorphic open-string vertex algebras and twisted modules from Courant algebroids
本文从带广义度量和联络的传递Courant代数胚构造亚纯开弦顶点代数及其扭曲模，实现Dirac算子和广义Hodge Laplacian的顶点算子表示，并推广到精确情形下的微分形式。
The Hurwitz space is the moduli space of pairs $(X,f)$ where $X$ is a compact Riemann surface and $f$ is a meromorphic function on $X$. We study the Laplace operator $Δ^{|df|^2}$ of the flat singular Riemannian manifold $(X,|df|^2)$. We define a regularized determinant for $Δ^{|df|^2}$ and study it as a functional on the Hurwitz space. We prove that this functional is related to a system of PDE which admits explicit integration. This leads to an explicit expression for the determinant of the Laplace operator in terms of the basic objects on the underlying Riemann surface (the prime form, theta…
Two characters on one punctured Riemann surface
本文提出了一种抽象框架，用于在穿孔黎曼面上实现两个多值微分，并通过极小长度最小化方法构建极小表面。
Extending the Labourie-Loftin correspondence, we establish, on any punctured oriented surface of finite type, a one-to-one correspondence between convex projective structures with specific types of ends and punctured Riemann surface structures endowed with meromorphic cubic differentials whose poles are at the punctures. This generalizes previous results of Loftin, Benoist-Hulin and Dumas-Wolf.
Let $M$ be an open Riemann surface. We prove that every meromorphic function on $M$ is the complex Gauss map of a conformal minimal immersion $M\to\mathbb{R}^3$ which may furthermore be chosen as the real part of a holomorphic null curve $M\to\mathbb{C}^3$. Analogous results are proved for conformal minimal immersions $M\to\mathbb{R}^n$ for any $n>3$. We also show that every conformal minimal immersion $M\to\mathbb{R}^n$ is isotopic through conformal minimal immersions $M\to\mathbb{R}^n$ to a flat one, and we identify the path connected components of the space of all conformal minimal immersio…
This work is based on the approach developed by J.~Dorfmeister, F.~Pedit and H.~Wu [GANG and KITCS preprint, Report KITCS94-4-1] to construct maps $Φ:D\rightarrow R^3$, $D$ being the unit disk in $C$, whose images are surfaces of constant mean curvature. They start from certain meromorphic one forms, so called meromorphic potentials, which take values in a twisted loop algebra over $su(2)$. The authors give necessary and sufficient conditions on the meromorphic potential for the constructed map $Φ$ to be an immersion. This revision corrects several misprints and minor TeX problems.
A Non-Abelian Approach to Riemann Surfaces Part I: Wronskian Geometry
针对黎曼曲面上的射影平坦全纯向量丛开展研究，通过为其分配Wronskian线丛、定义亚纯截面除法，将Abel恒等式解释为对应线丛的第一陈类。
Deformations of singularities of meromorphic $\mathfrak{sl}_2(\mathbb{C})$-connections and meromorphic quadratic differentials
本文研究无迹 $2\times2$ 亚纯线性 ODE 奇点的形变与汇合，建立形式不变量与二次微分射流的对应，并刻画等单值形变及其退化。
Finiteness of Fuchsian equations with unitarizable monodromy on compact Riemann surfaces
该研究证明紧黎曼曲面上特定亚纯射影联络的有限性，建立和乐的正则性与增长估计，推导球面锥形度量及稳定扩张分歧除子的有限性结果。
By constructing an ODE through a kind of meromorphic 1-forms, we will give an explicit construction of a kind of conformal metrics of constant curvature on Riemann surfaces with singularities. As an application, we will classify constant curvature one metrics on $S^{2}$ with two conical singularities, which was first proved by Troyanov in \cite{Tr89} by using projective connection.
On limit sets for geodesics of meromorphic connections
本文利用Fuchsian亚纯联络诱导的分支仿射结构，构造了具有无穷多自交点和奇特omega-极限集的测地线例子，填补了亚纯联络测地线渐近行为存在性研究的空白。
We give a new, connected-sum-like construction of Riemannian metrics with special holonomy G_2 on compact 7-manifolds. The construction is based on a gluing theorem for appropriate elliptic partial differential equations. As a prerequisite, we also obtain asymptotically cylindrical Riemannian manifolds with holonomy SU(3) building up on the work of Tian and Yau. Examples of new topological types of compact 7-manifolds with holonomy G_2 are constructed using Fano 3-folds.
Given any compact Riemann surface $C$, there is a canonical meromorphic 2--form $\widehatη$ on $C\times C$, with pole of order two on the diagonal $Δ\, \subset\, C\times C$, constructed in \cite{cfg}. This meromorphic 2--form $\widehatη$ produces a canonical projective structure on $C$. On the other hand the uniformization theorem provides another canonical projective structure on any compact Riemann surface $C$. We prove that these two projective structures differ in general. This is done by comparing the $(0,1)$--component of the differential of the corresponding sections of the moduli space…
Gauss-Bonnet formula for metrics with logarithmic singularities on compact Riemann surfaces
该研究在高斯曲率关于度量面积形式Lebesgue可积的条件下，推广了紧黎曼曲面上带对数奇异性度量的Gauss-Bonnet公式，并将其应用于关联三次微分亚纯的特殊Kähler度量。
We study the regularized determinant of the Laplacian as a functional on the space of Mandelstam diagrams (noncompact translation surfaces glued from finite and semi-infinite cylinders). A Mandelstam diagram can be considered as a compact Riemann surface equipped with a conformal flat singular metric $|ω|^2$, where $ω$ is a meromorphic one-form with simple poles such that all its periods are pure imaginary and all its residues are real. The main result is an explicit formula for the determinant of the Laplacian in terms of the basic objects on the underlying Riemann surface (the prime form, th…
A (meromorphic) quadratic differential is a (meromorphic) section of the tensor square of the canonical bundle of a Riemann surface. They arose in the study of quasiconformal mappings in the works of Oswald Teichmüller, and have played a mayor role in the study of the Riemann moduli, where they can be identified with cotangent vectors to the moduli space. In more recent years, inspired by several observations by Kontsevich, Soibelman and Seidel, in arXiv:1302.7030 Bridgeland and Smith identified certain spaces of meromorphic quadratic differentials with the space of stability conditions on a t…
Isomonodromic and isospectral deformations of meromorphic connections: the $\mathfrak{sl}_2(\mathbb{C})$ case
本文针对 $\mathfrak{sl}_2(\mathbb{C})$ 中的亚纯联络，给出几何规范下 Lax 对的显式表达式及 Darboux 坐标变换，连接等单演与等谱形变，解决先前遗留问题。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。