Kodai Mathematical Journal

On the multiplicity of the image of simple closed curves via holomorphic maps between compact Riemann surfaces

Hiroshi Yamamoto

Full-text: Open access

Abstract

Every non-trivial closed curve $C$ on a compact Riemann surface $R$ is freely homotopic to the $r$-fold iterate ${C_0}^r$ of some primitive closed geodesic $C_0$ on $R$. We call $r$ the multiplicity of $C$, and denote it by $N_{R} (C)$. Let $f$ be a non-constant holomorphic map of a compact Riemann surface $R_1$ of genus $g_1$ onto another compact Riemann surface $R_2$ of genus $g_2$ with $g_1 \geq g_2 > 1$, and $C$ a simple closed geodesic of hypebolic length $l_{R_1} (C)$ on $R_1$. In this paper, we give an upper bound for $N_{R_2} (f(C))$ depending only on $g_1$, $g_2$ and $l_{R_1} (C)$.

Article information

Source
Kodai Math. J., Volume 26, Number 1 (2003), 69-84.

Dates
First available in Project Euclid: 16 April 2003

Permanent link to this document
https://projecteuclid.org/euclid.kmj/1050496649

Digital Object Identifier
doi:10.2996/kmj/1050496649

Mathematical Reviews number (MathSciNet)
MR2004b:30077

Zentralblatt MATH identifier
1059.30033

Citation

Yamamoto, Hiroshi. On the multiplicity of the image of simple closed curves via holomorphic maps between compact Riemann surfaces. Kodai Math. J. 26 (2003), no. 1, 69--84. doi:10.2996/kmj/1050496649. https://projecteuclid.org/euclid.kmj/1050496649


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