Minimal submanifolds of Kähler–Einstein manifolds with equal Kähler angles

Isabel M.C. Salavessa and Giorgio Valli

Vol. 205 (2002), No. 1, 197–235

DOI: 10.2140/pjm.2002.205.197

Abstract

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≠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.

Milestones

Received: 10 April 2000

Revised: 9 March 2001

Published: 1 July 2002