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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The monotonicity of the ratio of two Abelian integrals


Authors: Changjian Liu and Dongmei Xiao
Journal: Trans. Amer. Math. Soc. 365 (2013), 5525-5544
MSC (2010): Primary 34C07, 34C08; Secondary 37G15
Published electronically: May 10, 2013
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Abstract: In this paper, we study the monotonicity of the ratio of two Abelian integrals

$\displaystyle I_0(h)=\int _{\Gamma _h}y\,dx\quad {\rm and}\quad I_1(h)=\int _{\Gamma _h}xy\,dx,$

where $ \Gamma _h$ is a compact component of the level set $ \{(x,y):\ y^2+\Psi (x)=h, \ h\in J\}$; here $ J$ is an open interval. We first give a new criterion for determining the monotonicity of the ratio of the above two Abelian integrals. Then using this new criterion, we obtain some new Hamiltonian functions $ H(x,y)$ so that the ratio of the associated two Abelian integrals is monotone. Especially when $ H(x,y)$ has the form $ y^2+P_5(x)$, we obtain the sufficient and necessary conditions that the ratio of two Abelian integrals is monotone, where $ P_5(x)$ is a polynomial of $ x$ with degree five.

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Additional Information

Changjian Liu
Affiliation: School of Mathematics, Soochow University, Suzhou 215006, People’s Republic of China
Email: liucj@suda.edu.cn

Dongmei Xiao
Affiliation: Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
Email: xiaodm@sjtu.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05934-X
PII: S 0002-9947(2013)05934-X
Keywords: Abelian integral, monotonicity, hyperelliptic Hamiltonian
Received by editor(s): April 1, 2012
Published electronically: May 10, 2013
Additional Notes: The first author was partially supported by the NSFC grant (No. 10901117) and pre-research of Soochow University
The second author was the corresponding author and was partially supported by the NSFC grants (No. 10831003 and No. 10925102) and the Program of Shanghai Subject Chief Scientists (No. 10XD1406200)
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.