Minimal submanifolds of Kähler-Einstein manifolds with equal Kähler angles

We consider F:M → N a minimal submanifold M of real dimension 2n, immersed into a Kähler–Einstein manifold N of complex dimension 2n, and scalar curvature R. We assume that n ≥ 2 and F has equal Kähler angles. Our main result is to prove that, if n = 2 and R 6 = 0, then F is either a complex submanifold or a Lagrangian submanifold. We also prove that, if n ≥ 3, M is compact and orientable, then: (A) If R < 0, then F is complex or Lagrangian; (B) If R = 0, the Kähler angle must be constant. We also study pluriminimal submanifolds with equal Kähler 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. 1. Introduction. Let (N, J, g) be a Kähler manifold of complex dimension 2n and F:M →