[Paper Review] Generalized Lagrangian mean curvature flows in symplectic manifolds

Knut Smoczyk, Mu‐Tao Wang|ArXiv.org|Oct 14, 2009

Geometry and complex manifolds4 references3 citations

TL;DR

This paper introduces a generalized Lagrangian mean curvature flow in almost Kähler manifolds using a metric and complex connection $\widehat{\nabla}$ with an Einstein condition on its Ricci form. The flow preserves the Lagrangian condition when $\widehat{\rho} = f\omega$, extending classical flows to broader symplectic settings, including cotangent bundles and almost Einstein Kähler manifolds.

ABSTRACT

An almost Kähler structure on a symplectic manifold $(N, ω)$ consists of a Riemannian metric $g$ and an almost complex structure $J$ such that the symplectic form $ω$ satisfies $ω(\cdot, \cdot)=g(J(\cdot), \cdot)$. Any symplectic manifold admits an almost Kähler structure and we refer to $(N, ω, g, J)$ as an almost Kähler manifold. In this article, we propose a natural evolution equation to investigate the deformation of Lagrangian submanifolds in almost Kähler manifolds. A metric and complex connection $\hn$ on $TN$ defines a generalized mean curvature vector field along any Lagrangian submanifold $M$ of $N$. We study the evolution of $M$ along this vector field, which turns out to be a Lagrangian deformation, as long as the connection $\hn$ satisfies an Einstein condition. This can be viewed as a generalization of the classical Lagrangian mean curvature flow in Kähler-Einstein manifolds where the connection $\hn$ is the Levi-Civita connection of $g$. Our result applies to the important case of Lagrangian submanifolds in a cotangent bundle equipped with the canonical almost Kähler structure and to other generalization of Lagrangian mean curvature flows, such as the flow considered by Behrndt \cite{b} in Kähler manifolds that are almost Einstein.

Motivation & Objective

• To define a generalized mean curvature flow for Lagrangian submanifolds in general almost Kähler manifolds, where the ambient space lacks a Kähler-Einstein structure.
• To overcome the failure of classical mean curvature flow to preserve Lagrangian conditions in general symplectic manifolds by introducing a generalized mean curvature vector via a connection $\widehat{\nabla}$.
• To establish conditions under which the generalized flow preserves the Lagrangian condition, extending known results to non-Kähler and non-Einstein settings.
• To demonstrate that the flow applies to important geometric examples, such as cotangent bundles with canonical almost Kähler structures and almost Einstein Kähler manifolds.
• To unify and generalize existing flows, including the classical Lagrangian mean curvature flow and Behrndt's modified flow, under a single framework.

Proposed method

• Define a generalized mean curvature vector $\vec{\widehat{H}}$ on almost Lagrangian submanifolds using a metric and complex connection $\widehat{\nabla}$ on the tangent bundle $TN$.
• Introduce the Einstein condition on $\widehat{\nabla}$, requiring its Ricci form $\widehat{\rho}$ to satisfy $\widehat{\rho} = f\omega$ for some smooth function $f$.
• Construct the generalized mean curvature flow as the evolution $\frac{\partial F}{\partial t} = \vec{\widehat{H}}$, with initial data $F(M,0) = M_0$.
• Use Cartan's formula and curvature identities to show that $\frac{\partial}{\partial t} F^*\omega = \Delta(F^*\omega) + \text{lower-order terms}$, ensuring the Lagrangian condition is preserved.
• Apply parabolic regularity and short-time existence theory to establish smooth solutions for compact initial Lagrangian submanifolds.
• Utilize the closedness of the generalized mean curvature 1-form $\widehat{H}$ and the Einstein condition to control the evolution of $F^*\omega$.

Experimental results

Research questions

• RQ1 Under what conditions on a connection $\widehat{\nabla}$ on an almost Kähler manifold does the generalized mean curvature flow preserve the Lagrangian condition?
• RQ2 Can the classical Lagrangian mean curvature flow in Kähler-Einstein manifolds be recovered as a special case of this generalized framework?
• RQ3 Does the generalized flow extend to non-Kähler symplectic manifolds such as cotangent bundles equipped with the canonical almost Kähler structure?
• RQ4 How does the torsion of $\widehat{\nabla}$ affect the generalized mean curvature vector and the preservation of the Lagrangian condition?
• RQ5 Can the framework accommodate flows previously studied in almost Einstein Kähler manifolds, such as those of Behrndt?

Key findings

• The generalized mean curvature flow exists smoothly for any compact initial almost Lagrangian submanifold in an almost Kähler manifold, with maximal time $T \in (0, \infty]$.
• The flow preserves the Lagrangian condition if and only if the connection $\widehat{\nabla}$ satisfies the Einstein condition $\widehat{\rho} = f\omega$.
• In the case of Kähler-Einstein manifolds, choosing $\widehat{\nabla}$ as the Levi-Civita connection recovers the classical Lagrangian mean curvature flow.
• For almost Einstein Kähler manifolds, the generalized flow includes Behrndt's modified flow as a special case when $\widehat{\nabla} = \nabla + \sigma \otimes J$ with $\sigma = d^c\psi$.
• On the cotangent bundle $T^*M$ of a Riemannian manifold $M$, the canonical connection $\widehat{\nabla}$ is Einstein with $f = 0$, so the flow preserves Lagrangian submanifolds.
• The evolution of $F^*\omega$ satisfies a parabolic inequality $\frac{\partial}{\partial t}|F^*\omega|^2 \leq \Delta|F^*\omega|^2 + c|F^*\omega|^2$, ensuring that $F^*\omega \equiv 0$ for all time if $M_0$ is Lagrangian.