# Conformal hyperbolic metrics with mixed cone and cusp singularities

Let $X$ be a compact Riemann surface and $D=\sum_{j=1}^n\,(\theta_j-1)\,P_j$ be a ${\Bbb R}$-divisor on $X$ such that $\theta_j\geq 0$ and $P_1,\cdots,P_n$ are $n$ distinct points on $X$. We call $ds^2$ a conformal metric representing $D$ if $ds^2$ is a smooth conformal metric on $X\backslash {\rm Supp}\, D:=X\backslash \{P_1,\cdots, P_n\}$ and in a neighborhood $U_j$ of $P_j$, $ds^2$ has form $e^{2u_j}\,|dz|^2$, where $z$ is a local complex coordinate defined in $U_j$ centered at $P_j$, as $\theta_j>0$ the real valued function $u_j-(\theta_j-1)\,\ln\,|z|$ is continuous in $U_j$, and as $\theta_j=0$ the real valued function $u_j+\ln\,|z|+\ln\,\big(-\ln\,|z|\big)$ is continuous in $U_j$. We also call that $ds^2$ has cone singularity of angle $2\pi\theta_j$ at $P_j$ as $\theta_j>0$, and has cusp singularity at $P_j$ as $\theta_j=0$. Note that $ds^2$ has finite area near a cone or cusp singularity. It is well known that if a conformal flat or spherical (positive constant curvature) metric has finite area, then its isolated singularities must be cone singularities. The Uniformization Theory gives a class of conformal hyperbolic metrics on Riemann surfaces with mixed cusp singularities and cone singularities of angles in $\{2\pi/2,\,2\pi/3,\, 2\pi/4,\cdots\}$. By the Gauss-Bonnet formula, if $ds^2$ is a conformal hyperbolic metric representing $D=\sum_{j=1}^n\,(\theta_j-1)\,P_j$, then by Gauss-Bonnet formula there holds $\chi(X)+\sum_{j=1}^n\,(\theta_j-1)<0$ and such a metric exists uniquely by the maximum principle. Both McOwen and Troyanov used PDE to show that there exists a unique conformal hyperbolic metric on $X$ representing a ${\Bbb R}$-divisor $D=\sum_{j=1}^n\,(\theta_j-1)\,P_j$ with $\theta_j>0$ if and only if the above inequality holds. I would like to ask whether the condition of $\theta_j>0$ could be relaxed to $\theta_j\geq 0$ there. That is, is the following statement true or false?

$\bullet$ There exists a conformal hyperbolic metric representing a ${\Bbb R}$-divisor $D=\sum_{j=1}^n\,(\theta_j-1)\,P_j$ with $\theta_j\geq 0$ on a compact Riemann surface $X$ if $\chi(X)+\sum_{j=1}^n\,(\theta_j-1)<0$.

I believe it is correct and have been pondering over a proof for some days.

## 2 Answers

This is correct, and the same proof as in McOwen and Troyanov should work. In fact they were not the first who proved this result. The story begins with E. Picard, who wrote several papers on this (also using PDE methods), and the paper of M. Heins:

MR0143901 Heins, Maurice On a class of conformal metrics. Nagoya Math. J. 21 1962 1–60. (Chapter II, The Problem of Schwarz-Picard).

EDIT. I confess that I have not read all papers of Picard on the subject, but according to Heins, Picard proved the statement for $\theta_j>0$. So I believe that Heins was the first to prove the statement with $\theta_j\geq 0$, as you ask.

The method used in the papers my McOwen and Troyanov you cite do not directly extend to the case of cusps. The paper of M. Heins cited by A. Eremenko does (if I read it correctly), it is based on Perron's method. The result you desire is also proved in the paper "Sur la courbure des surfaces ouvertes" (D. Hulin and M. Troyanov). A more complete investigation is in "Prescribing curvature on open surfaces" (same authors). These papers are here http://sma.epfl.ch/~troyanov/Papers/CRAS_1990.pdf http://sma.epfl.ch/~troyanov/Papers/MathAnnalen92.pdf A. Eremenko correctly recall that the problem was first investigated by Picard around 1900.