Maslov class and minimality in Calabi-Yau manifolds

TL;DR

This paper extends the concept of the Maslov class to Lagrangian submanifolds in Calabi-Yau manifolds, linking it to the mean curvature and suggesting broader generalizations in symplectic geometry.

Contribution

It provides a consistent definition of the Maslov class in Calabi-Yau settings and relates it to geometric quantities like the mean curvature vector.

Findings

Maslov class defined for Lagrangian submanifolds in Calabi-Yau manifolds

Maslov class represented by contraction of Kaehler form with mean curvature

Proposed generalization of Maslov class for broader symplectic manifolds

Abstract

Generalizing the construction of the Maslov class for a Lagrangian embedding in a symplectic vector space, we prove that it is possible to give a consistent definition of this class for any Lagrangian submanifold of a Calabi-Yau manifold. Moreover, we prove that this class can be represented by the contraction of the Kaehler form associated to the Calabi-Yau metric, with the mean curvature vector field of the Lagrangian embedding. Finally, we suggest a possible generalization of the Maslov class for Lagrangian submanifolds of any symplectic manifold, via the mean curvature representation.