Abstract
It has been known for some time that there exist 5 essentially different real forms of the complex affine Kac–Moody algebra of type \(A_2^{(2)}\) and that one can associate 4 of these real forms with certain classes of “integrable surfaces,” such as minimal Lagrangian surfaces in \(\mathbb {CP}^2\) and \(\mathbb {CH}^2\), as well as definite and indefinite affine spheres in \({\mathbb {R}}^3\). In this paper, we consider the class of timelike minimal Lagrangian surfaces in the indefinite complex hyperbolic two-space \(\mathbb {CH}^{2}_1\). We show that this class of surfaces corresponds to the fifth real form. Moreover, for each timelike Lagrangian surface in \(\mathbb {CH}^{2}_1\) we define natural Gauss maps into certain homogeneous spaces and prove a Ruh–Vilms-type theorem, characterizing timelike minimal Lagrangian surfaces among all timelike Lagrangian surfaces in terms of the harmonicity of these Gauss maps.
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References
Anciaux, H.: Minimal Submanifolds in Pseudo-Riemannian Geometry. World Scientific Publishing Co. Pte. Ltd., Hackensack (2011)
Baird, P., Wood, J.C.: Harmonic Morphisms Between Riemannian Manifolds. London Mathematical Society Monographs. New Series. The Clarendon Press, Oxford University Press, Oxford (2003)
Back-Valente, V., Bardy-Panse, N., Ben Messaoud, H., Rousseau, G.: Formes presque-déployées des algèbres de Kac–Moody: classification et racines relatives. J. Algebra 171(1), 43–96 (1995)
Ben Messaoud, H., Rousseau, G.: Classification des formes réelles presque compactes des algèbres de Kac–Moody affines. J. Algebra 267(2), 443–513 (2003)
Boothby, W.M., Wang, H.C.: On contact manifolds. Ann. Math. 2(68), 721–734 (1958)
Dillen, F., Vrancken, L.: Lorentzian isotropic Lagrangian immersions. Filomat 30(10), 2857–2867 (2016)
Dorfmeister, J., Eitner, U.: Weierstraß-type representation of affine spheres. Abh. Math. Sem. Univ. Hamburg 71, 225–250 (2001)
Dorfmeister, J.F., Freyn, W., Kobayashi, S.-P., Wang, E.: Survey on real forms of the complex \(A_2^{(2)}\)-Toda equation. Complex Manifolds 6, 194–227 (2019). arXiv:1902.01558
Dorfmeister, J.F., Inoguchi, J., Kobayashi, S.-P.: Constant mean curvature surfaces in hyperbolic 3-space via loop group. J. Reine Angew. Math. 686, 1–36 (2014)
Dorfmeister, J., Inoguchi, J., Toda, M.: Weierstraß type representations of timelike surfaces with constant mean curvature. Integr Syst Diff Geometry Contemp. Math. 308, 77–99 (2002)
Dorfmeister, J.F., Ma, H.: Minimal lagrangian surfaces in \({{\mathbb{C}}} P^2\) via the loop group method Part I: the contractible case (2020). arXiv:2002.01318
Dorfmeister, J.F., Wang, E.: Definite affine spheres via loop groups I: general theory, In preparation, (2019)
Heintze, E., Groß, C.: Finite order automorphisms and real forms of affine Kac–Moody algebras in the smooth and algebraic category. Mem. Am. Math. Soc. 219, 1030 (2012)
Inoguchi, J., Toda, M.: Timelike minimal surfaces via loop groups. Acta Appl. Math. 83(3), 313–355 (2004)
Inoguchi, J., Udagawa, S.: Affine spheres and finite gap solutions of Tzitzèica equation. J. Phys. Commun. 2(11), 115020 (2018)
Kac, V.: Infinite-dimensional Lie algebras, 3rd edn. Cambridge University Press, Cambridge (1990)
Kulkarni, R.S.: An analogue of the Riemann mapping theorem for Lorentz metrics. Proc. Roy. Soc. Lond. Ser. A 401, 117–130 (1985)
Kobayashi, S.-P.: Real forms of complex surfaces of constant mean curvature. Trans. Am. Math. Soc. 363(4), 1765–1788 (2011)
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, Vol. II. Reprint of the 1969 Original. A Wiley-Interscience Publication. Wiley, New York (1996)
Ma, H., Ma, Y.: Totally real minimal tori in \({\mathbb{C}}{\rm P}^2\). Math. Z. 249(2), 241–267 (2005)
Loftin, J., McIntosh, I.: Minimal Lagrangian surfaces in \(\mathbb{CH}^2\) and representations of surface groups into \(SU(2,1)\). Geom. Dedicata 162, 67–93 (2013)
Loftin, J., McIntosh, I.: Cubic differentials in the differential geometry of surfaces. IRMA Lect. Math. Theor. Phys., 27: Handbook of Teichmüller theory. Vol. VI, 231–274, (2016)
McIntosh, I.: Special Lagrangian cones in \({{\mathbb{C}}^3}\) and primitive harmonic maps. J. Lond. Math. Soc. (2) 67(3), 769–789 (2003)
Reckziegel, H.: Horizontal lifts of isometric immersions into the bundle space of a pseudo-Riemannian submersion. Glob. Diff. Geometry Glob. Anal. Springer Lectures Notes 1156, 264–279 (1984)
Ruh, A.E., Vilms, J.: The tension field of the Gauss map. Trans. Am. Math. Soc. 149, 569–573 (1970)
Shapiro, R.A.: Pseudo-Hermitian symmetric spaces. Comment. Math. Helv. 46, 529–548 (1971)
Weinstein, T.: An Introduction to Lorentz Surfaces, De Gruyter Expositions in Mathematics 22. Walter de Gruyter & Co., Berlin (1996)
Yuxin, D.: On indefinite special Lagrangian submanifolds in indefinite complex Euclidean spaces. J. Geom. Phys. 59(6), 710–726 (2009)
Acknowledgements
We would like to thank Hui Ma for pointing out to us (5) in Remark 1.12. We would also like to thank the referee for carefully reading the manuscript and for pointing out to us a number of typographical errors in the manuscript.
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Shimpei Kobayashi is partially supported by JSPS KAKENHI Grant Number JP18K03265 and Deutsche Forschungsgemeinschaft-Collaborative Research Center, TRR 109, “Discretization in Geometry and Dynamics”.
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Dorfmeister, J.F., Kobayashi, S. Timelike minimal Lagrangian surfaces in the indefinite complex hyperbolic two-space. Annali di Matematica 200, 521–546 (2021). https://doi.org/10.1007/s10231-020-01005-1
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DOI: https://doi.org/10.1007/s10231-020-01005-1