Abstract
A two-dimensional periodic Schrödingier operator is associated with every Lagrangian torus in the complex projective plane \({\mathbb C}P^2\). Using this operator, we introduce an energy functional on the set of Lagrangian tori. It turns out this energy functional coincides with the Willmore functional \(W^{-}\) introduced by Montiel and Urbano. We study the energy functional on a family of Hamiltonian-minimal Lagrangian tori and support the Montiel–Urbano conjecture that the minimum of the functional is achieved by the Clifford torus. We also study deformations of minimal Lagrangian tori and show that if a deformation preserves the conformal type of the torus, then it also preserves the area, i.e., preserves the value of the energy functional. In particular, the deformations generated by Novikov–Veselov equations preserve the area of minimal Lagrangian tori.
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References
Carberry, E., McIntosh, I.: Minimal Lagrangian 2-tori in \({\mathbb{C}}P^2\) come in real families of every dimension. J. Lond. Math. Soc. (2) 69(2), 531–544 (2004)
Castro, I., Urbano, F.: New examples of minimal Lagrangian tori in the complex projective plane. Manuscr. Math. 85, 265–281 (1994)
Dorfmeister, J.F., Ma, H.: Explicit expressions for the Iwasawafactors, the metric and the monodromy matrices for minimal Lagrangian surfaces in \(\mathbb{C}P^2\). In: Dynamical Systems, Numbertheory and Applications, pp. 19–47. World Science Publishing, Hackensack (2016)
Dorfmeister, J.F., Ma, H.: A new look at equivariant minimal Lagrangian surfaces in \(\mathbb{C}P^2\). In: Geometry and Topology of Manifolds, pp. 97–125. Springer Proc. Math. Stat., vol. 154. Springer, Berlin (2016)
Haskins, H.: The geometric complexity of special Lagrangian \(T^2\)-cones. Invent. Math. 157(1), 11–70 (2004)
Joyce, D.: Special Lagrangian submanifolds with isolated conical singularities. v. Survey and applications. J. Differ. Geom. 63(2), 279–347 (2003)
Ma, H., Ma, Y.: Totally real minimal tori in \(\mathbb{C}P^2\). Math. Z. 249(2), 241–267 (2005)
Ma, H.: Hamiltonian stationary Lagrangian surfaces in \(\mathbb{C}P^2\). Ann. Glob. Anal. Geom. 27, 1–16 (2005)
Ma, H., Schmies, M.: Examples of Hamiltonian stationary Lagrangian tori in \(\mathbb{C}P^2\). Geom. Dedicata 118, 173–183 (2006)
Marques, F.C., Neves, A.: Min–max theory and the Willmore conjecture. Ann. Math. 2(179), 683–782 (2014)
Mironov, A.E.: The Novikov–Veselov hierarchy of equations and integrable deformations of minimal Lagrangian tori in \(\mathbb{C}P^2\). (Russian) Sib. Elektron. Mat. Izv. 1, 38–46 (also arXiv:math/0607700v1) (2004)
Mironov, A.E.: New examples of Hamilton-minimal and minimal Lagrangian submanifolds in \({\mathbb{C}}^n\) and \({\mathbb{C}}P^n.\). Sb. Math. 195(1), 85–96 (2004)
Mironov, A.E.: Relationship between symmetries of the Tzitzéica equation and the Novikov–Veselov hierarchy. Math. Notes 82(4), 569–572 (2007)
Montiel, S., Urbano, F.: A Willmore functional for compact surfaces in the complex projective plane. J. Reine Angew. Math. 546, 139–154 (2002)
Oh, Y.: Volume minimization of Lagrangian submanifolds under Hamiltonian deformations. Math. Z. 212, 175–192 (1993)
Sharipov, R.A.: Minimal tori in the five-dimensional sphere in \({\mathbb{C}}^3\). Theor. Math. Phys. 87(1), 363–369 (1991)
Taimanov, I.A.: Two-dimensional Dirac operator and the theory of surfaces. Rus. Math. Surv. 61(1), 79–159 (2006)
Taimanov, I.A.: Modified Novikov–Veselov equation and differential geometry of surfaces. Am. Math. Soc. Transl. Ser. 2 179, 133–151 (1997)
Veselov, A.P., Novikov, S.P.: Finite-gap two-dimensional potential Schrödinger operators. Explicit formulas and evolution equations (Russian). Dokl. Akad. Nauk SSSR 279(1), 20–24 (1984)
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The authors are grateful to professor Yong Luo and professor Iskander Taimanov for their useful discussions.
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Hui Ma was supported in part by NSFC Grant No. 11271213. Andrey E. Mironov was supported by the Russian Foundation for Basic Research (Grant 16-51-55012) and by a Grant from Dmitri Zimin’s Dynasty foundation. Dafeng Zuo was supported in part by NSFC (Grant Nos. 11671371, 11371338) and the Fundamental Research Funds for the Central Universities.
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Ma, H., Mironov, A.E. & Zuo, D. An energy functional for Lagrangian tori in \(\mathbb {C}P^2\) . Ann Glob Anal Geom 53, 583–595 (2018). https://doi.org/10.1007/s10455-017-9589-6
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DOI: https://doi.org/10.1007/s10455-017-9589-6