We build analytic surfaces in represented by the most general sixth Painlevé equation PVI in two steps. First, the moving frame of the surfaces built by Bonnet in 1867 is extrapolated to a new, second order, isomonodromic matrix Lax pair of PVI, whose elements depend rationally on the dependent variable and quadratically on the monodromy exponents θj. Second, by converting back this Lax pair to a moving frame, we obtain an extrapolation of Bonnet surfaces to surfaces with two more degrees of freedom. Finally, we give a rigorous derivation of the quantum correspondence for PVI.
REFERENCES
1.
Abramowitz
, M.
and Stegun
, I.
, Handbook of Mathematical Functions, Tenth Printing
(Dover
, New York
, 1972
).2.
Babich
, M. V.
and Bordag
, L. A.
, “Projective differential geometrical structure of the Painlevé equations
,” J. Differ. Equations
157
, 452
–485
(1999
).3.
Bobenko
, A. I.
, “Surfaces in terms of 2 by 2 matrices: Old and new integrable cases
,” in Harmonic Maps and Integrable Systems
, Aspects of Mathematics E23, edited by Fordy
, A. P.
and Wood
, J. C.
(Vieweg
, Braunschweig, Wiesbaden
, 1994
), pp. 83
–128
.4.
Bobenko
, A. I.
and Eitner
, U.
, “Bonnet surfaces and Painlevé equations
,” J. Reine Angew. Math.
1998
(499
), 47
–79
.5.
Bobenko
, A. I.
and Eitner
, U.
, Painlevé Equations in Differential Geometry of Surfaces
, Lecture Notes in Mathematics 1753 (Springer
, Berlin
, 2000
), p. 120
.6.
Bobenko
, A. I.
, Eitner
, U.
, and Kitaev
, A. V.
, “Surfaces with harmonic inverse mean curvature and Painlevé equations
,” Geom. Dedicata
68
, 187
–227
(1997
).7.
Bonnet
, O.
, “Mémoire sur la théorie des surfaces applicables sur une surface donnée. Deuxième partie: Détermination de toutes les surfaces applicables sur une surface donnée
,” J. Ec. Polytech.
42
, 1
–151
(1867
).8.
Cartan
, É.
, “Sur les couples de surfaces applicables avec conservation des courbures principales
,” Bull. Sci. Math.
66
, 55
–72
(1942
), 74–85.9.
Chazy
, J.
, “Sur les équations différentielles du troisième ordre et d’ordre supérieur dont l’intégrale générale a ses points critiques fixes
,” Acta Math.
34
, 317
–385
(1911
).10.
Chen
, W.
and Li
, H.
, “Bonnet surfaces and isothermic surfaces
,” Results Math.
31
, 40
–52
(1997
).11.
Conte
, R.
, “On the Lax pairs of the sixth Painlevé equation
,” RIMS Kôkyûroku Bessatsu
B2
, 21
–27
(2007
); e-print arXiv:nlin/0701049.12.
Conte
, R.
, “Surfaces de Bonnet et équations de Painlevé
,” C. R. Math. Acad. Sci. Paris
355
, 40
–44
(2017
); e-print arXiv:1607.01222 [math-ph].13.
Conte
, R.
and Dornic
, I.
, “The master Painlevé VI heat equation
,” C. R. Acad. Sci. Paris
352
, 803
–806
(2014
); e-print arXiv:1409.1166 [math-ph].14.
Conte
, R.
, Grundland
, A. M.
, and Musette
, M.
, “A reduction of the resonant three-wave interaction to the generic sixth Painlevé equation
,” J. Phys. A: Math. Gen.
39
, 12115
–12127
(2006
); e-print arXiv:nlin/0604011, Special issue One hundred years of Painlevé VI.15.
(Regular and Chaotic Dynamics, Moscow, 2011) (in Russian)].
Conte
, R.
and Musette
, M.
, The Painlevé Handbook
(Springer
, Berlin
, 2008
) [ (Regular and Chaotic Dynamics, Moscow, 2011) (in Russian)].16.
17.
Fuchs
, R.
, “Sur quelques équations différentielles linéaires du second ordre
,” C. R. Acad. Sci.
141
, 555
–558
(1905
).18.
Garnier
, R.
, “Sur des équations différentielles du troisième ordre dont l’intégrale générale est uniforme et sur une classe d’équations nouvelles d’ordre supérieur dont l’intégrale générale a ses points critiques fixes
,” Ann. Sci. Ec. Norm. Super.
29
, 1
–126
(1912
).19.
Garnier
, R.
, “Sur un théorème de Schwarz
,” Comment. Math. Helvetici
25
, 140
–172
(1951
).20.
Halphen
, G.-H.
, Traité des fonctions elliptiques et de leurs applications
, Première partie, Théorie des Fonctions Elliptiques et de Leurs Développements en Série (Gauthier-Villars
, Paris
, 1886
), p. 492
, http://gallica.bnf.fr/document?O=N007348.21.
Harnad
, J.
, “Dual isomonodromic deformations and moment maps to loop algebras
,” Commun. Math. Phys.
166
, 337
–365
(1994
).22.
Hazzidakis
, J. N.
, “Biegung mit erhaltung der hauptkrümmungsradien
,” J. Reine Angew. Math.
1897
(117
), 42
–56
.23.
Heun
, K.
, “Zur theorie der Riemann’schen functionen zweiter Ordnung mit vier verzweigungspunkten
,” Math. Ann.
33
, 161
–179
(1889
).24.
Jimbo
, M.
and Miwa
, T.
, “Monodromy preserving deformations of linear ordinary differential equations with rational coefficients. II
,” Phys. D
2
, 407
–448
(1981
).25.
Jimbo
, M.
and Sakai
, H.
, “A q-analog of the sixth Painlevé equation
,” Lett. Math. Phys.
38
, 145
–154
(1996
).26.
Lin
, R.
, Conte
, R.
, and Musette
, M.
, “On the Lax pairs of the continuous and discrete sixth Painlevé equations
,” J. Nonlinear Math. Phys.
10
(Suppl. 2
), 107
–118
(2003
).27.
Mahoux
, G.
, “Introduction to the theory of isomonodromic deformations of linear ordinary differential equations with rational coefficients
,” in The Painlevé Property, One Century Later
, CRM Series in Mathematical Physics, edited by Conte
, R.
(Springer
, New York
, 1999
), pp. 35
–76
.28.
Malmquist
, J.
, “Sur les équations différentielles du second ordre dont l’intégrale générale a ses points critiques fixes
,” Arkiv Math. Astr. Fys.
17
, 1
–89
(1922–1923
).29.
Manin
, Yu. I.
, “Sixth Painlevé equation, universal elliptic curve and mirror of P2
,” in Geometry of Differential Equations
, AMS Translations: Series 2, edited by Khovanskii
, A.
, Varchenko
, A.
, and Vassiliev
, V.
(American Mathematical Society Translations
, 1998
), 186(39), pp. 131
–151
; e-print arXiv:alg-geom/9605010.30.
Novikov
, D. P.
, “The 2×2 matrix Schlesinger system and the Belavin-Polyakov-Zamolodchikov system
,” Teor. Mat. Fiz.
161
, 191
–203
(2009
)Novikov
, D. P.
, [Theor. Math. Phys.
161
, 1485
–1496
(2009
)]. 31.
Noumi
, M.
and Yamada
, Y.
, “A new Lax pair for the sixth Painlevé equation associated with so(8)
,” in Microlocal Analysis and Complex Fourier Analysis
, edited by Fujita
, K.
and Kawai
, T.
(World Scientific
, Singapore
, 2002
), pp. 238
–252
; e-print arXiv:math-ph/0203029.32.
Okamoto
, K.
, “Polynomial Hamiltonians associated with Painlevé equations. II. Differential equations satisfied by polynomial Hamiltonians
,” Proc. Jpn. Acad., Ser. A
56
, 367
–371
(1980
).33.
Painlevé
, P.
, “Sur les équations différentielles du second ordre et d’ordre supérieur dont l’intégrale générale est uniforme
,” Acta Math.
25
, 1
–85
(1902
).34.
Painlevé
, P.
, “Sur les équations différentielles du second ordre à points critiques fixes
,” C. R. Acad. Sci. Paris
143
, 1111
–1117
(1906
).35.
Phillips
, E. R.
, “Karl M. Peterson: The earliest derivation of the Mainardi-Codazzi equations and the fundamental theorem of surface theory
,” Hist. Math.
6
, 137
–163
(1979
).36.
Picard
, É.
, “Mémoire sur la théorie des fonctions algébriques de deux variables
,” J. Math. Pure Appl.
5
, 135
–319
(1889
).37.
Poincaré
, H.
, “Sur les groupes des équations linéaires
,” Acta Math.
4
, 201
–312
(1883
);38.
Heun’s Differential Equations
, edited by Ronveaux
, A.
(Oxford University Press
, Oxford
, 1995
).39.
Schlesinger
, L.
, “Über eine Klasse von differentialsystemen beliebiger Ordnung mit festen kritischen Punkten
,” J. Reine Angew. Math.
1912
(141
), 96
–145
.40.
de Sparre
, C.
, “Sur l’équation … premier mémoire
,” Acta Math.
3
, 105
–140
(1883
) [“Deuxième mémoire,” Acta Math. 3, 289–321 (1883)].41.
Springborn
, B. A.
, “Bonnet pairs in the 3-sphere
,” Contemp. Math.
308
, 297
–303
(2002
).42.
Suleimanov
, B. I.
, “Hamiltonian property of the Painlevé equations and the method of isomonodromic deformations
,” Differ. Uravn.
30
, 791
–796
(1994
) [Differ. Equations 30, 726–732 (1994)].43.
Suleimanov
, B. I.
, “‘Quantum’ linearization of Painlevé equations as a component of their L, A pairs
,” Ufa Math. J.
4
(2
), 127
–136
(2012
), ISSN 2304-0122; e-print arXiv:1302.6716.44.
Tsegel’nik
, V. V.
, “Hamiltonians associated with the sixth Painlevé equation
,” Teor. Mat. Fiz.
151
, 54
–65
(2007
) Tsegel’nik
, V. V.
, [Theor. Math. Phys.
151
, 482
–491
(2007
)].45.
Veselov
, A. P.
, “On Darboux-Treibich-Verdier potentials
,” Lett. Math. Phys.
96
, 209
–216
(2011
); e-print arXiv:1004.5355 [math-ph].46.
Voss
, K.
, Bonnet Surfaces in Spaces of Constant Curvature
, Lecture Notes, First MSJ International Research on Geometry and Global Analysis (Research institute Sendai
, Japan
, 1993
), pp. 295
–307
.47.
Zabrodin
, A.
and Zotov
, A.
, “Quantum Painlevé-Calogero correspondence
,” J. Math. Phys.
53
, 073507
(2012
); e-print arXiv:1107.5672.48.
Zabrodin
, A.
and Zotov
, A.
, “Quantum Painlevé-Calogero correspondence for Painlevé VI
,” J. Math. Phys.
53
, 073508
(2012
); e-print arXiv:1107.5672.49.
Zotov
, A.
, “Elliptic linear problem for Calogero-Inozemtsev model and Painlevé VI equation
,” Lett. Math. Phys.
67, 153
–165
(2004
); e-print arXiv:hep-th/0310260.50.
The Codazzi equations were in fact first written in 1853 by
Peterson
, K. M.
, a Latvian student, before Mainardi, 1856, and Codazzi, 1868, see the historical notes by Phillips.35 51.
Erratum. In formula 18.6.23 of Abramowitz and Stegun1, the last g2 should be g3.
52.
This means that both surfaces have the same first fundamental form.
