Communications in Analysis and Geometry

Volume 28 (2020)

Number 3

From Lagrangian to totally real geometry: coupled flows and calibrations

Pages: 607 – 675



Jason D. Lotay (Mathematical Institute, University of Oxford, United Kingdom)

Tommaso Pacini (Dipartimento di Matematica, Universita’ di Torino, Italy)


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-Kähler 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, further studied in [18], 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 Kähler manifolds. We also discuss possible applications to Lagrangian submanifolds and calibrated geometry.

Received 29 December 2016

Accepted 13 December 2017

Published 6 July 2020