论文标题
球体和代数数的同轴指标
Co-Axial Metrics on the Sphere and Algebraic Numbers
论文作者
论文摘要
在本文中，我们考虑以下曲率方程$$Δu+{\ rm e}^u =4π\ bigGl（（（θ_0-1））Δ_0+（θ_1-1）Δ_1+\ sum_ {j = 1} \ text {in} \ \ mathbb r^2，$$ $ $ $ u（x）= -2（1+θ_\ infty）\ ln | x |+o（1）\ qquad \ qquad \ text {as} \ | x | x | x | x | x | \ to \ to 其中$θ_0$，$θ_1$，$θ_\ infty $，而$θ_{j}'$是积极的非智能者，$ 1 \ le j \ le j \ le n $，而$θ_{j} {j}'\ in \ mathbb {n} n} _ {n} _ {\ geq 2} $ n是$ n $ n $ n+le;从几何上讲，解决方案$ u $产生圆锥公制$ {\ rm d} s^2 = \ frac12 {\ rm e}^u | {\ rm d} x |^x |^2 $ curvature $ 1 $在球体上，带有圆锥形的奇异性，$ 0 $，$ 1 $，$，$，$，$，$，$，$ t_ $ t_和t_ $ t_ $ t_和t_ $ t_和t_ n+m $，带有角度的$2πθ_0$，$2πθ_1$，$2πθ_\ infty $和$2πθ_{J}'$ at $ 0 $，$ 1 $，$ \ infty $和$ t_j $。如果有一个开发的地图$ h（x）$ $ u $，则公制$ {\ rm d} s^2 $或解决方案$ u $称为同轴，这是由Mondello和Panov介绍的，因此投影型单模型组包含在单位圈中。 Mondello-Panov（2016）和Eremenko（2020）获得了这种指标存在的足够和必要条件。在本文中，我们固定角度并研究奇异性的位置$ t_1，\ dots，t_ {n+m} $。令$ a \ subset \ mathbb {c}^{n+m} $为这些$（t_1，\ dots，t_ {n+m}）$的集合，因此存在共同轴向的度量，包括$ m = 1 $，即$ 1 $ $ queger' $ a $是有限套装。此外，对于$ n = 0 $的情况，我们获得了集合$ a $的基数的锐利界限。由于Eremenko，Gabrielov和Tarasov（2016）以及超几何方程的单构型，我们应用了结果，以获得这种结合。 （ii）如果$ m \ ge 2 $，则$ a $是代数的尺寸$ \ leq m-1 $。
In this paper, we consider the following curvature equation $$Δu+{\rm e}^u=4π\biggl((θ_0-1)δ_0+(θ_1-1)δ_1 +\sum_{j=1}^{n+m}\bigl(θ_j'-1\bigr)δ_{t_j}\biggr)\qquad \text{in}\ \mathbb R^2,$$ $$u(x)=-2(1+θ_\infty)\ln|x|+O(1)\qquad \text{as} \ |x|\to\infty,$$ where $θ_0$, $θ_1$, $θ_\infty$, and $θ_{j}'$ are positive non-integers for $1\le j\le n$, while $θ_{j}'\in\mathbb{N}_{\geq 2}$ 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 $2πθ_0$, $2πθ_1$, $2πθ_\infty$, and $2πθ_{j}'$ 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 $θ_{n+1}'$ among $θ_j'$, 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 $\leq m-1$.
