Maslov, Chern-Weil and Mean Curvature

TL;DR

This paper derives an integral formula for the Maslov index of a pair of bundles over a surface, linking it to curvature and boundary data, with applications to Kähler manifolds and totally real submanifolds.

Contribution

It introduces a new integral formula for the Maslov index involving curvature and boundary terms, extending Chern-Weil theory to boundary cases.

Findings

Provides bounds on the Maslov index based on geometric conditions.

Establishes monotonicity results for the Maslov index in specific settings.

Connects the Maslov index to curvature and boundary geometry in Kähler manifolds.

Abstract

We provide an integral formula for the Maslov index of a pair $(E,F)$ over a surface $\Sigma$, where $E\rightarrow\Sigma$ is a complex vector bundle and $F\subset E_{|\partial\Sigma}$ is a totally real subbundle. As in Chern-Weil theory, this formula is written in terms of the curvature of $E$ plus a boundary contribution. When $(E,F)$ is obtained via an immersion of $(\Sigma,\partial\Sigma)$ into a pair $(M,L)$ where $M$ is K\"ahler and $L$ is totally real, the formula allows us to control the Maslov index in terms of the geometry of $(M,L)$. We exhibit natural conditions on $(M,L)$ which lead to bounds and monotonicity results.