An algebraic construction of solutions to the mean field equations on hyperelliptic curves and its adiabatic limit

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Abstract

In this paper, we give an algebraic construction of the solution to the following mean field equation
$$
\Delta \psi+e^{\psi}=4\pi\sum_{i=1}^{2g+2}\delta_{P_{i}},
$$
on a genus $g\geq 2$ hyperelliptic curve $(X,ds^{2})$ where $ds^{2}$ is a canonical metric on $X$ and $\{P_{1},\cdots,P_{2g+2}\}$ is the set of Weierstrass points on $X.$
Original languageEnglish
Pages (from-to)3693
Number of pages3707
JournalProceedings of the American Mathematical Society
Volume146
Publication statusPublished - 2018 Jul

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Weierstrass Point
Mean Field Equation
Hyperelliptic Curves
Pi
Genus
Metric

Cite this

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title = "An algebraic construction of solutions to the mean field equations on hyperelliptic curves and its adiabatic limit",
abstract = "In this paper, we give an algebraic construction of the solution to the following mean field equation$$\Delta \psi+e^{\psi}=4\pi\sum_{i=1}^{2g+2}\delta_{P_{i}},$$on a genus $g\geq 2$ hyperelliptic curve $(X,ds^{2})$ where $ds^{2}$ is a canonical metric on $X$ and $\{P_{1},\cdots,P_{2g+2}\}$ is the set of Weierstrass points on $X.$",
author = "Jia-Ming Liou and Chih-Chung Liu",
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journal = "Proceedings of the American Mathematical Society",
issn = "0002-9939",
publisher = "American Mathematical Society",

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AU - Liu, Chih-Chung

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N2 - In this paper, we give an algebraic construction of the solution to the following mean field equation$$\Delta \psi+e^{\psi}=4\pi\sum_{i=1}^{2g+2}\delta_{P_{i}},$$on a genus $g\geq 2$ hyperelliptic curve $(X,ds^{2})$ where $ds^{2}$ is a canonical metric on $X$ and $\{P_{1},\cdots,P_{2g+2}\}$ is the set of Weierstrass points on $X.$

AB - In this paper, we give an algebraic construction of the solution to the following mean field equation$$\Delta \psi+e^{\psi}=4\pi\sum_{i=1}^{2g+2}\delta_{P_{i}},$$on a genus $g\geq 2$ hyperelliptic curve $(X,ds^{2})$ where $ds^{2}$ is a canonical metric on $X$ and $\{P_{1},\cdots,P_{2g+2}\}$ is the set of Weierstrass points on $X.$

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JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

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