Maslov, Chern-Weil and Mean Curvature

We provide an integral formula for the Maslov index of a pair $(E,F)$ over a surface $\mathrm{\Sigma }$, where $E\to \mathrm{\Sigma }$ is a complex vector bundle and $F\subset E_{\mathrm{\mid }\mathrm{\partial }\mathrm{\Sigma }}$ 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 $(\mathrm{\Sigma },\mathrm{\partial }\mathrm{\Sigma })$ 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.