Abstract
An interesting class of submanifolds of a Kähler manifold M2n is the class of submanifolds Nn ⊑ M2n which are minimal with respect to the metric on M2n and are Lagrangian with respect to the symplectic form on M2n. A general Kähler manifold will not have any of these submanifolds. However, in this paper, we show that if the metric on M2n is also Einstein, then these minimal Lagrangian submanifolds exist in abundance, at least locally. We give a precise description of this "generality" in terms of Cartan-Kähler theory and relate these submanifolds to the calibrated geometries of Harvey and Lawson and to maximal real structures on algebraic varieties.
Keywords
- Complex Manifold
- Differential System
- Algebraic Variety
- Lagrangian Submanifold
- Frame Field
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Bibliography
S.S. Chern, Complex Manifolds without Potential Theory, 2nd edition, Springer-Verlag, 1979
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Dennis de Turck and Jerry Kazdan, Some Regularity Theorems in Riemannian Geometry, Ann. Scient. Ec. Norm. Sup., 4e série, t. 14, 1981, pp 249–260
Reese Harvey and Blaine Lawson, Calibrated Geometries, Acta Mathematica, v. 148 (1982), pp 47–157
Alan Weinstein, Lectures on Symplectic Manifolds, CMBS Series in Mathematics, no. 29, AMS, 1977
S.T. Yau, Survey on Partial Differential Equations in Differential Geometry, Annals of Math. Studies, no. 102, Princeton University Press, 1982
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© 1987 Springer-Verlag
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Bryant, R.L. (1987). Minimal lagrangian submanifolds of Kähler-einstein manifolds. In: Gu, C., Berger, M., Bryant, R.L. (eds) Differential Geometry and Differential Equations. Lecture Notes in Mathematics, vol 1255. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0077676
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DOI: https://doi.org/10.1007/BFb0077676
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-17849-1
Online ISBN: 978-3-540-47883-6
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