Abstract
We show that the Green functions on flat tori can have either three or five critical points only. There does not seem to be any direct method to attack this problem. Instead, we have to employ sophisticated nonlinear partial differential equations to study it. We also study the distribution of the number of critical points over the moduli space of flat tori through deformations. The functional equations of special theta values provide important inequalities which lead to a solution for all rhombus tori.
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[CCL] S. A. Chang, C. Chen, and C. Lin, "Extremal functions for a mean field equation in two dimension," in Lectures on Partial Differential Equations, Int. Press, Somerville, MA, 2003, vol. 2, pp. 61-93.
@incollection {CCL, MRKEY = {2055839},
AUTHOR = {Chang, Sun-Yung A. and Chen, Chiun-Chuan and Lin, Chang-Shou},
TITLE = {Extremal functions for a mean field equation in two dimension},
BOOKTITLE = {Lectures on Partial Differential Equations},
SERIES = {New Stud. Adv. Math.},
VOLUME = {2},
PAGES = {61--93},
PUBLISHER = {Int. Press, Somerville, MA},
YEAR = {2003},
MRCLASS = {35J60 (35J20 49J10)},
MRNUMBER = {2005d:35061},
MRREVIEWER = {Anna Maria Rossi},
ZBLNUMBER = {1071.35040},
} -
[Chen-Lin1]
C. Chen and C. Lin, "Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces," Comm. Pure Appl. Math., vol. 55, iss. 6, pp. 728-771, 2002.@article {Chen-Lin1, MRKEY = {1885666},
AUTHOR = {Chen, Chiun-Chuan and Lin, Chang-Shou},
TITLE = {Sharp estimates for solutions of multi-bubbles in compact {R}iemann surfaces},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {55},
YEAR = {2002},
NUMBER = {6},
PAGES = {728--771},
ISSN = {0010-3640},
CODEN = {CPAMA},
MRCLASS = {53C21 (35J60 58E12)},
MRNUMBER = {2003d:53056},
MRREVIEWER = {Tonia Ricciardi},
DOI = {10.1002/cpa.3014},
ZBLNUMBER = {1040.53046},
} -
[Chen-Lin2] C. Chen and C. Lin, "Topological degree for a mean field equation on Riemann surfaces," Comm. Pure Appl. Math., vol. LVI, pp. 1667-1727, 2003.
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author = {Chen, Chiun-Chuan and Lin, Chang-Shou},
TITLE = {Topological degree for a mean field equation on Riemann surfaces},
JOURNAL={Comm. Pure Appl. Math.},
VOLUME={LVI},
YEAR={2003},
PAGES={1667--1727},
MRNUMBER={2004h:35065},
ZBLNUMBER={1032.58010},
} -
[Chen-Lin3] C. Chen and C. Lin, Topological degree for a mean field equation with singular data.
@misc{Chen-Lin3,
author = {Chen, Chiun-Chuan and Lin, Chang-Shou},
TITLE = {Topological degree for a mean field equation with singular data},
NOTE={in preparation},
} -
[CLW] C. Chen, C. Lin, and G. Wang, "Concentration phenomena of two-vortex solutions in a Chern-Simons model," Ann. Sc. Norm. Super. Pisa Cl. Sci., vol. 3, iss. 2, pp. 367-397, 2004.
@article {CLW, MRKEY = {2075988},
AUTHOR = {Chen, Chiun-Chuan and Lin, Chang-Shou and Wang, Guofang},
TITLE = {Concentration phenomena of two-vortex solutions in a {C}hern-{S}imons model},
JOURNAL = {Ann. Sc. Norm. Super. Pisa Cl. Sci.},
FJOURNAL = {Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V},
VOLUME = {3},
YEAR = {2004},
NUMBER = {2},
PAGES = {367--397},
ISSN = {0391-173X},
MRCLASS = {58E50 (35J60 81T13)},
MRNUMBER = {2005m:58033},
MRREVIEWER = {Pierpaolo Esposito},
ZBLNUMBER = {1170.35413},
} -
[JLW] J. Jost, C. Lin, and G. Wang, "Analytic aspects of the Toda system. II. Bubbling behavior and existence of solutions," Comm. Pure Appl. Math., vol. 59, iss. 4, pp. 526-558, 2006.
@article {JLW, MRKEY = {2199785},
AUTHOR = {Jost, J{ü}rgen and Lin, Chang-Shou and Wang, Guofang},
TITLE = {Analytic aspects of the {T}oda system. {II}. {B}ubbling behavior and existence of solutions},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {59},
YEAR = {2006},
NUMBER = {4},
PAGES = {526--558},
ISSN = {0010-3640},
CODEN = {CPAMA},
MRCLASS = {35J60 (35B45 35J45 47J30)},
MRNUMBER = {2007h:35099},
MRREVIEWER = {Gabriella Tarantello},
DOI = {10.1002/cpa.20099},
ZBLNUMBER={pre05023695},
} -
[Li] Y. Y. Li, "Harnack type inequality: the method of moving planes," Comm. Math. Phys., vol. 200, iss. 2, pp. 421-444, 1999.
@article {Li, MRKEY = {1673972},
AUTHOR = {Li, Yan Yan},
TITLE = {Harnack type inequality: the method of moving planes},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {200},
YEAR = {1999},
NUMBER = {2},
PAGES = {421--444},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {58E15 (53C21)},
MRNUMBER = {2000c:58024},
MRREVIEWER = {Olivier Druet},
DOI = {10.1007/s002200050536},
ZBLNUMBER = {0928.35057},
} -
[LW] C. -S. Lin and C. -L. Wang, On the minimality of extra critical points of Green functions on flat tori.
@misc{LW,
author={Lin, C.-S. and Wang, C.-L.},
TITLE={On the minimality of extra critical points of Green functions on flat tori},
NOTE={in preparation},
} -
[NT0] M. Nolasco and G. Tarantello, "On a sharp Sobolev-type inequality on two-dimensional compact manifolds," Arch. Ration. Mech. Anal., vol. 145, iss. 2, pp. 161-195, 1998.
@article {NT0, MRKEY = {1664542},
AUTHOR = {Nolasco, Margherita and Tarantello, Gabriella},
TITLE = {On a sharp {S}obolev-type inequality on two-dimensional compact manifolds},
JOURNAL = {Arch. Ration. Mech. Anal.},
FJOURNAL = {Archive for Rational Mechanics and Analysis},
VOLUME = {145},
YEAR = {1998},
NUMBER = {2},
PAGES = {161--195},
ISSN = {0003-9527},
MRCLASS = {58E15 (26D10 46E35 58J05)},
MRNUMBER = {2000j:58021},
MRREVIEWER = {Nguy{ê}{ñ} H{ô}{\`n}g Th{á}i},
DOI = {10.1007/s002050050127},
ZBLNUMBER = {0980.46022},
} -
[NT] M. Nolasco and G. Tarantello, "Double vortex condensates in the Chern-Simons-Higgs theory," Calc. Var. Partial Differential Equations, vol. 9, iss. 1, pp. 31-94, 1999.
@article {NT, MRKEY = {1710938},
AUTHOR = {Nolasco, Margherita and Tarantello, Gabriella},
TITLE = {Double vortex condensates in the {C}hern-{S}imons-{H}iggs theory},
JOURNAL = {Calc. Var. Partial Differential Equations},
FJOURNAL = {Calculus of Variations and Partial Differential Equations},
VOLUME = {9},
YEAR = {1999},
NUMBER = {1},
PAGES = {31--94},
ISSN = {0944-2669},
MRCLASS = {58E50 (35B32 35Q40 81T13)},
MRNUMBER = {2001e:58022},
DOI = {10.1007/s005260050132},
ZBLNUMBER = {0951.58030},
} -
[NT1] M. Nolasco and G. Tarantello, "Vortex condensates for the ${ SU}(3)$ Chern-Simons theory," Comm. Math. Phys., vol. 213, iss. 3, pp. 599-639, 2000.
@article {NT1, MRKEY = {1785431},
AUTHOR = {Nolasco, Margherita and Tarantello, Gabriella},
TITLE = {Vortex condensates for the {${\rm SU}(3)$} {C}hern-{S}imons theory},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {213},
YEAR = {2000},
NUMBER = {3},
PAGES = {599--639},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {58E15 (81T13)},
MRNUMBER = {2001j:58025},
MRREVIEWER = {Yisong Yang},
DOI = {10.1007/s002200000252},
ZBLNUMBER = {0998.81047},
} -
[Whittaker] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge: Cambridge Univ. Press, 1996.
@book {Whittaker, MRKEY = {1424469},
AUTHOR = {Whittaker, E. T. and Watson, G. N.},
TITLE = {A Course of Modern Analysis},
SERIES = {Cambridge Mathematical Library},
NOTE = {reprint of the fourth (1927) edition},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {1996},
PAGES = {vi+608},
ISBN = {0-521-58807-3},
MRCLASS = {01A75 (30-01 33-01)},
MRNUMBER = {97k:01072},
ZBLNUMBER = {0951.30002},
}
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