Minimal submanifolds of Kaehler-Einstein manifolds with equal Kaehler angles

Details Minimal submanifolds of Kaehler-Einstein manifolds with equal Kaehler angles Minimal submanifolds of Kaehler-Einstein manifolds with equal Kaehler angles

2000-02-07

arxiv.org/abs/math/0002050v2

Mathematics

Differential Geometry

33 pages, plain LaTeX, minor revisions

Scientific paper

We consider $F: M \to N$ a minimal oriented compact real 2n-submanifold M, immersed into a Kaehler-Einstein manifold N of complex dimension 2n, and scalar curvature R. We assume that $n \geq 2$ and F has equal Kaehler angles. Our main result is to prove that, if n = 2 and $R \neq 0$, then F is either a complex submanifold or a Lagrangian submanifold. We also prove that, if $n \geq 3$ and F has no complex points, then: (A) If R < 0, then F is Lagrangian; (B) If R = 0, the Kaehler angle must be constant. We also study pluriminimal submanifolds with equal Kaehler angles, and prove that, if they are not complex submanifolds, N must be Ricci-flat and there is a natural parallel homothetic isomorphism between TM and the normal bundle.

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