Geodesic
Veselov-Novikov hierarchy of equations, and integrable deformations of minimal Lagrangian tori in~$\mathbb CP^2$
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 1 (2004), pp. 38-46.

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We associate a periodic two-dimensional Schrödinger operator to every Lagrangian torus in $\mathbb CP^2$ and define the spectral curve of a torus as the Floquet spectrum on this operator on the zero energy level. In this event minimal Lagrangian tori correspond to potential operators. We show that the Novikov–Veselov hierarchy of equations induces integrable deformations of a minimal Lagrangian torus in $\mathbb CP^2$ preserving the spectral curve. 
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A. E. Mironov. Veselov-Novikov hierarchy of equations, and integrable deformations of minimal Lagrangian tori in~$\mathbb CP^2$. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 1 (2004), pp. 38-46. http://geodesic.mathdoc.fr/item/SEMR_2004_1_a3/

[1] A. P. Veselov, S. P. Novikov, “Konechnozonnye dvumernye potentsialnye operator Shredingera. Yavnye formuly i evolyutsionnye uravneniya”, DAN CCCP, 279:1 (1984), 20–24 | MR | Zbl

[2] I. A. Taimanov, “Predstavlenie Veiershtrassa zamknutykh poverkhnostei v $\mathbb R^3$”, Funktsion. analiz i ego pril., 32:4 (1998), 49–62 | MR | Zbl

[3] B. A. Dubrovin, S. P. Novikov, I. M. Krichever, “Uravneniya Shredingera v periodicheskom pole i rimanovy poverkhnosti”, DAN SSSR, 229:1 (1976), 15–18 | MR | Zbl

[4] A. E. Mironov, “O novykh primerakh gamiltonovo–minimalnykh i minimalnakh lagranzhevykh podmnogoobraziyakh v $\mathbb C^n$ i $\mathbb CP^n$”, Matem. sb., 195:1 (2004), 89–102 | MR | Zbl

[5] I. Castro, F. Urbano, “Examples of unstable Hamiltonian-minimal Lagrangian tori in $\mathbb C^2$”, Compositio Mathematica, 111 (1998), 1–14 | DOI | MR | Zbl

[6] D. Joyce, “Special Lagrangian 3-folds and integrable systems”, Surveys on geometry and integrable systems, Adv. Stud. Pure Math., 51, Math. Soc. Japan, Tokyo, 2008, 189–233 | MR | Zbl

[7] R. A. Sharipov, “Minimalnye tory v pyatimernoi sfere”, Teor. i matem. fizika, 87:1 (1991), 48–56 | MR | Zbl

[8] H. Ma, J. Ma, Totally real minimal tori in $CP^2$, arXiv: math.DG/0106141

[9] E. Carberry, I. Mcintosh, “Minimal lagrangian 2-tori in $\mathbb CP^2$ come in real families of every dimention”, London J. of Math., 69:2 (2004), 531–544 | DOI | MR | Zbl

[10] N. Hitchin, “Harmonic maps from a 2-torus to the 3-sphere”, J. Diff. Geom., 31 (1990), 627–710 | MR | Zbl

[11] A. V. Mikhailov, “The reduction problem and the scattering method”, Physica 3D, 1 (1981), 73–117 | Zbl

[12] B. A. Dubrovin, V. B. Matveev, S. P. Novikov, “Nelineinye uravneniya tipa Kortvega–de Friza, konechnozonnye lineinye operatory i abelevy mnogoobraziya”, Uspekhi matem. nauk, 31:1 (1976), 55–136 | MR | Zbl

[13] B. G. Konopelchenko, “Induced surfaces and their integrable dynamics”, Stud. Appl. Math., 96:1 (1996), 9–51 | MR | Zbl

[14] I. A. Taimanov, “Modified Novikov–Veselov equation and differential geometry of surfaces”, Amer. Math. Soc. Transl., Ser. 2, 179 (1997), 133–151 | MR | Zbl

[15] A. E. Mironov, “O gamiltonovo-minimalnykh lagranzhevykh torakh v $\mathbb CP^2$”, Sibirskii matem. zhurnal, 44:6 (2003), 1324–1328 | MR | Zbl

[16] I. M. Krichever, “Nelineinye uravneniya i ellipticheskie krivye”, Sovremennye problemy matematiki, VINITI, 23, 1983, 79–136 | MR | Zbl