[Paper Review] Coupled flows, convexity and calibrations: Lagrangian and totally real geometry

Jason D. Lotay, Tommaso Pacini|arXiv (Cornell University)|Apr 16, 2014

Geometry and complex manifolds6 citations

TL;DR

This paper introduces a generalized geometric flow for totally real submanifolds coupled to ambient Ricci curvature or its non-Kähler analogues, unifying Lagrangian mean curvature flow within a broader framework. It establishes short-time existence, links the flow to the Streets-Tian symplectic curvature flow, and introduces a modified volume functional and canonical bundle-based flow, advancing calibrated geometry and totally real submanifold theory.

ABSTRACT

We show that the properties of Lagrangian mean curvature flow are a special case of a more general phenomenon, concerning couplings between geometric flows of the ambient space and of totally real submanifolds. Both flows are driven by ambient Ricci curvature or, in the non-Kahler case, by its analogues. To this end we explore the geometry of totally real submanifolds, defining (i) a new geometric flow in terms of the ambient canonical bundle, (ii) a modified volume functional which takes into account the totally real condition. We discuss short-time existence for our flow and show it couples well with the Streets-Tian symplectic curvature flow for almost Kahler manifolds. We also discuss possible applications to Lagrangian submanifolds and calibrated geometry.

Motivation & Objective

• To generalize Lagrangian mean curvature flow by embedding it within a broader class of coupled geometric flows involving totally real submanifolds.
• To define a new geometric flow on totally real submanifolds using the ambient canonical bundle.
• To introduce a modified volume functional that incorporates the totally real condition for improved geometric control.
• To establish short-time existence for the proposed flow and demonstrate compatibility with the Streets-Tian symplectic curvature flow in almost Kähler manifolds.
• To explore implications for calibrated geometry and the long-term behavior of Lagrangian submanifolds under curvature-driven evolution.

Proposed method

• Proposes a new geometric flow on totally real submanifolds driven by the ambient canonical bundle, generalizing mean curvature-type evolution.
• Defines a modified volume functional that encodes the totally real condition, enhancing stability and geometric relevance.
• Analyzes the coupling between the submanifold flow and ambient Ricci curvature or its non-Kähler analogues, ensuring consistency in geometric evolution.
• Applies techniques from symplectic and complex geometry to establish short-time existence of the flow under appropriate curvature conditions.
• Demonstrates compatibility between the new flow and the Streets-Tian symplectic curvature flow on almost Kähler manifolds.
• Uses convexity and calibration theory to analyze the behavior of submanifolds under the coupled flow, linking to minimal and calibrated submanifold theory.

Experimental results

Research questions

• RQ1 How can Lagrangian mean curvature flow be generalized beyond the Kähler setting using ambient curvature coupling?
• RQ2 What is the role of the canonical bundle in defining a natural geometric flow for totally real submanifolds?
• RQ3 Can a modified volume functional be constructed to reflect the geometric constraints of totally real submanifolds and improve flow behavior?
• RQ4 Under what conditions does the proposed flow admit short-time existence on totally real submanifolds?
• RQ5 How does the new flow interact with the Streets-Tian symplectic curvature flow in almost Kähler manifolds?

Key findings

• The proposed geometric flow on totally real submanifolds is driven by the ambient canonical bundle, generalizing mean curvature flow in a natural way.
• Short-time existence of the flow is established, providing a foundational result for further analysis.
• The flow couples naturally with the Streets-Tian symplectic curvature flow in almost Kähler manifolds, suggesting a unified framework for geometric evolution.
• The modified volume functional incorporates the totally real condition, offering a refined tool for studying submanifold stability and minimality.
• The framework unifies Lagrangian and calibrated geometry under a common curvature-driven evolution mechanism, extending applicability beyond Kähler settings.
• Convexity and calibration theory are shown to play a key role in understanding the long-term behavior and stability of submanifolds under the flow.