Abstract
The multiplier spectral curve of a conformal torus f : T 2 → S 4 in the 4-sphere is essentially (Bohle et al., Conformal maps from a 2-torus to the 4-sphere. arXiv:0712.2311) given by all Darboux transforms of f. In the particular case when the conformal immersion is a Hamiltonian stationary torus \({f: T^2 \to\mathbb{R}^4}\) in Euclidean 4-space, the left normal N : M → S 2 of f is harmonic, hence we can associate a second Riemann surface: the eigenline spectral curve of N, as defined in Hitchin (J Differ Geom 31(3):627–710, 1990). We show that the multiplier spectral curve of a Hamiltonian stationary torus and the eigenline spectral curve of its left normal are biholomorphic Riemann surfaces of genus zero. Moreover, we prove that all Darboux transforms, which arise from generic points on the spectral curve, are Hamiltonian stationary whereas we also provide examples of Darboux transforms which are not even Lagrangian.
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References
Anciaux H.: Construction of many Hamiltonian stationary Lagrangian surfaces in Euclidean four-space. Calc. Var. Partial Diff. Equ. 17(2), 105–120 (2003)
Bernstein H.: Non-special, non-canal isothermic tori with spherical lines of curvature. Trans. Am. Math. Soc. 353, 2245–2274 (2001)
Bohle, C., Leschke, K., Pedit, F., Pinkall, U.: Conformal maps from a 2-torus to the 4-sphere. arXiv:0712.2311
Bohle C., Peters G., Pinkall U.: Constrained Willmore surfaces. Calc. Var. Partial Diff. Equ. 32(2), 263–277 (2008)
Burstall, F., Ferus, D., Leschke, K., Pedit, F., Pinkall, U.: Conformal geometry of surfaces in S 4 and quaternions. In: Lecture Notes in Mathematics, Springer, Berlin, Heidelberg (2002)
Carberry, E., Leschke, K., Pedit, F.: Darboux transforms and spectral curves of constant mean curvature surfaces revisited (in preparation)
Castro I., Chen B.-Y.: Lagrangian surfaces in complex Euclidean plane via spherical and hyperbolic curves. Tohoku Math. J., 2nd ser 58(4), 565–579 (2006)
Castro I., Urbano F.: Examples of unstable Hamiltonian-minimal Lagrangian tori in \({\mathbb{C}^2}\) . Compositio Mathematica 111, 1–14 (1998)
Chen B.-Y.: Construction of Lagrangian surfaces in complex Euclidean plane with Legendre curves. Kodai Math. J. 29(1), 84–112 (2006)
Chen B.-Y., Morvan J.-M.: Deformations of isotropic submanifolds in Kähler manifolds. J. Geom. Phys. 13, 79–104 (1994)
Darboux G.: Sur les surfaces isothermiques. C. R. Acad. Sci. Paris 128, 1299–1305 (1899)
Ferus D., Leschke K., Pedit F., Pinkall U.: Quaternionic holomorphic geometry: Plücker formula, Dirac eigenvalue estimates and energy estimates of harmonic 2-tori. Invent. Math. 146, 507–593 (2001)
Grinevich P., Schmidt M.: Conformal invariant functionals of immersions of tori in \({\mathbb{R}^3}\) . J. Geom. Phys. 26, 51–78 (1998)
Hélein F., Romon P.: Weierstrass representation of Lagrangian surfaces in four dimensional space using spinors and quaternions. Commentarii Mathematici Helvetici 75, 668–680 (2000)
Hélein F., Romon P.: Hamiltonian stationary Lagrangian surfaces in \({\mathbb{C}^2}\) . Commun. Anal. Geom. 10(1), 79–126 (2002)
Hitchin N.: Harmonic maps from a 2-torus to the 3-sphere. J. Differ. Geom. 31(3), 627–710 (1990)
Leschke, K.: Transformation on Willmore surfaces. Habilitationsschrift, Universität Augsburg (2006)
McIntosh, I., Romon, P.: The spectral data for Hamiltonian stationary Lagrangian tori in \({\mathbb{R}^4}\) . arXiv:0707.1767
Moriya, K.: Hamiltonian stationary Lagrangian tori in the complex Euclidean plane with rational spectral curve. arXiv:0710.4233
Oh Y.-G.: Volume minimization of Lagrangian submanifolds under Hamiltonian deformations. Math. Z. 212, 175–192 (1993)
Taimanov I.: The Weierstrass representation of closed surfaces in \({\mathbb{R}^3}\) . Funct. Anal. Appl. 32, 49–62 (1998)
Uhlenbeck K.: Harmonic maps into Lie groups (classical solutions of the chiral model). J. Diff. Geom. 30, 1–50 (1989)
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Katrin Leschke supported by DFG SPP 1154 “Global Differential Geometry”.
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Leschke, K., Romon, P. Darboux transforms and spectral curves of Hamiltonian stationary Lagrangian tori. Calc. Var. 38, 45–74 (2010). https://doi.org/10.1007/s00526-009-0278-6
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DOI: https://doi.org/10.1007/s00526-009-0278-6