An algebraic construction of a solution to the mean field equations on hyperelliptic curves and its adiabatic limit
Authors:
Jia-Ming (Frank) Liou and Chih-Chung Liu
Journal:
Proc. Amer. Math. Soc. 146 (2018), 3693-3707
MSC (2010):
Primary 14H55, 35J15
DOI:
https://doi.org/10.1090/proc/14054
Published electronically:
May 15, 2018
MathSciNet review:
3825825
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Abstract | References | Similar Articles | Additional Information
Abstract: In this paper, we give an algebraic construction of the solution to the following mean field equation: \begin{equation*} \Delta \psi +e^{\psi }=4\pi \sum _{i=1}^{2g+2}\delta _{P_{i}} \end{equation*} 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 $X.$
- Jerry L. Kazdan and F. W. Warner, Curvature functions for compact $2$-manifolds, Ann. of Math. (2) 99 (1974), 14–47. MR 343205, DOI https://doi.org/10.2307/1971012
- Jia-Ming Liou, Explicit solutions to the mean field equations on hyperelliptic curves of genus two, Differential Geom. Appl. 56 (2018), 173–186. MR 3759361, DOI https://doi.org/10.1016/j.difgeo.2017.08.005
- Rick Miranda, Algebraic curves and Riemann surfaces, Graduate Studies in Mathematics, vol. 5, American Mathematical Society, Providence, RI, 1995. MR 1326604
- David Mumford, Tata lectures on theta. II, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2007. Jacobian theta functions and differential equations; With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura; Reprint of the 1984 original. MR 2307768
- I. R. Shafarevich, On some infinite-dimensional groups, Rend. Mat. e Appl. (5) 25 (1966), no. 1-2, 208–212. MR 485898
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Additional Information
Jia-Ming (Frank) Liou
Affiliation:
Department of Mathematics, National Cheng Kung University, Taiwan — and — NCTS, Mathematics
Email:
fjmliou@mail.ncku.edu.tw
Chih-Chung Liu
Affiliation:
Department of Mathematics, National Cheng Kung University, Taiwan
MR Author ID:
1063388
Email:
cliu@mail.ncku.edu.tw
Received by editor(s):
June 29, 2017
Published electronically:
May 15, 2018
Communicated by:
Lei Ni
Article copyright:
© Copyright 2018
American Mathematical Society