Maslov, Chern–Weil and mean curvature

We provide an integral formula for the Maslov index of a pair (E,F) over a surface Σ, where E→Σ is a complex vector bundle and F⊂E|∂Σ 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 (Σ,∂Σ) into a pair (M,L) where M is Kähler and Lis totally real, the formula allows us to control the Maslov index in termsof the geometry of (M,L). We exhibit natural conditions on (M,L) which lead to bounds and monotonicity results.