Abstract
Given an almost para-Kähler manifold equipped with a metric and para-complex connection, we define a generalized second fundamental form and generalized mean curvature vector of space-like Lagrangian submanifolds. We then show that the deformation induced by this variant of the mean curvature vector field preserves the Lagrangian condition, if the connection satisfies also some Einstein condition. In case the almost para-Kähler structure is integrable, the flow coincides with the classical mean curvature flow in the pseudo-Riemannian context.
This is a preview of subscription content, access via your institution.
References
Alekseevsky D.V., Medori C., Tomassini A.: Homogeneous para-Kähler Einstein manifolds. Russ. Math. Surv. 64, 1–43 (2009)
Behrndt, T.: Lagrangian mean curvature flow in almost Kähler-Einstein manifolds. In: Proceedings of the Conference “Complex and Differential Geometry”, Hannover 2009. arXiv:0812.4256v2 (2009, accepted)
Cruceanu V., Fortuny P., Gadea P.M.: A survey on paracomplex geometry. Rocky Mt. J. Math. 26, 83–115 (1996)
Datta B., Subramanian S.: Nonexistence of almost complex structures on products of even-dimensional spheres. Topol. Appl. 36(1), 39–42 (1990). doi:10.1016/0166-8641(90)90034-Y
Ecker K.: Interior estimates and longtime solutions for mean curvature flow of noncompact spacelike hypersurfaces in Minkowski space. J. Differ. Geom. 45, 481–498 (1997)
Ecker K., Huisken G.: Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes. Commun. Math. Phys. 135, 595–613 (1991)
Etayo F., Santamaría R., Trías U.R.: The geometry of a bi-Lagrangian manifold. Differ. Geom. Appl. 24(1), 33–59 (2006). doi:10.1016/j.difgeo.2005.07.002
Gadea P.M., Montesinos Amilibia A.: Spaces of constant para-holomorphic sectional curvature. Pac. J. Math. 136(1), 85–101 (1989)
Huang, R.L.: Lagrangian mean curvature flow in Pseudo-Euclidean space. arXiv:0908.3070 (2009)
Ivanov S., Zamkovoy S.: Parahermitian and paraquaternionic manifolds. Differ. Geom. Appl. 23(2), 205–234 (2005). doi:10.1016/j.difgeo.2005.06.002
Kaneyuki S.: On classification of para-Hermitian symmetric spaces. Tokyo J. Math. 8(2), 473–482 (1985)
Kaneyuki S., Kozai M.: Paracomplex structures and affine symmetric spaces. Tokyo J. Math. 8(1), 81–98 (1985)
Li, G., Salavessa, I.M.C.: Mean curvature flow of spacelike graphs. arXiv:0804.0783 v4 (2008)
Smoczyk, K.: A canonical way to deform a Lagrangian submanifold. arXiv:dg-ga/9605005 (1996)
Smoczyk, K., Wang, M.-T.: Generalized Lagrangian mean curvature flows in symplectic manifolds. arXiv:0910.2667v1 (2009)
Vaisman, I.: Symplectic Geometry and Secondary Characteristic Classes. Progress in Mathematics, vol. 72. Birkhäuser Boston Inc., Boston (1987)
Xin, Y.: Minimal Submanifolds and Related Topics. Nankai Tracts in Mathematics, vol. 8. World Scientific Publishing Co. Inc., River Edge (2003)
Yano, K., Ishihara, S.: Tangent and Cotangent Bundles: Differential Geometry. Pure and Applied Mathematics, No. 16. Marcel Dekker Inc., New York (1973)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J. Jost.
This work was partially written in the framework of the Graduiertenkolleg 1463 “Analysis, Geometry and String Theory”. Part of it forms part of the first author’s doctoral thesis.
Rights and permissions
About this article
Cite this article
Chursin, M., Schäfer, L. & Smoczyk, K. Mean curvature flow of space-like Lagrangian submanifolds in almost para-Kähler manifolds. Calc. Var. 41, 111–125 (2011). https://doi.org/10.1007/s00526-010-0355-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00526-010-0355-x
Mathematics Subject Classification (2000)
- Primary: 53C44