Broadly-Pluriminimal Submanifolds of Kaehler-Einstein Manifolds

TL;DR

This paper introduces the concept of broadly-pluriminimal submanifolds in Kaehler-Einstein manifolds and characterizes their geometric properties under certain curvature and orientation conditions.

Contribution

It generalizes known results for minimal surfaces to higher dimensions, providing new classifications of submanifolds based on Kaehler angles and curvature.

Findings

If R < 0, submanifold has complex or Lagrangian directions.

For n=2, compact, no complex directions, then Lagrangian.

If constant Kaehler angles, not Lagrangian, then R=0.

Abstract

We define broadly-pluriminimal immersed 2n-submanifold F: M --> N into a Kaehler-Einstein manifold of complex dimension 2n and scalar curvature R. We prove that, if M is compact, n \geq 2, and R < 0, then: (i) Either F has complex or Lagrangian directions; (ii) If n = 2, M is oriented, and F has no complex directions, then it is a Lagrangian submanifold, generalising the well-known case n = 1 for minimal surfaces due to Wolfson. We also prove that, if F has constant Kaehler angles with no complex directions, and is not Lagrangian, then R = 0 must hold. Our main tool is a formula on the Laplacian of a symmetric function on the Kaehler angles.