Generalized Lagrangian mean curvature flow in K\"ahler manifolds that are almost Einstein

TL;DR

This paper introduces a generalized mean curvature flow for Lagrangian submanifolds in almost Einstein K"ahler manifolds, showing it preserves the Lagrangian condition and relates to the Lagrangian angle in special cases.

Contribution

It defines the notion of almost Einstein K"ahler manifolds and establishes a generalized mean curvature flow that preserves Lagrangian submanifolds, linking it to the Lagrangian angle.

Findings

Lagrangian submanifolds remain Lagrangian under the generalized flow.

In almost Einstein K"ahler manifolds with trivial canonical bundle, the flow relates to the Lagrangian angle.

The generalized mean curvature vector field is dual to the Lagrangian angle in special cases.

Abstract

We introduce the notion of K\"ahler manifolds that are almost Einstein and we define a generalized mean curvature vector field along submanifolds in them. We prove that Lagrangian submanifolds remain Lagrangian, when deformed in direction of the generalized mean curvature vector field. For a K\"ahler manifold that is almost Einstein, and which in addition has a trivial canonical bundle, we show that the generalized mean curvature vector field of a Lagrangian submanifold is the dual vector field associated to the Lagrangian angle.