Structure of a Morse form foliation on a closed surface in terms of genus

AbstractWe study the geometry of compact singular leaves γ and minimal components Cmin of the foliation Fω of a Morse form ω on a genus g closed surface Mg2 in terms of genus g(⁎). We show that c(ω)+∑γg(V(γ))+g(⋃Cmin¯)=g, where c(ω) is the number of homologically independent compact leaves and V(⁎) is a small closed tubular neighborhood. This allows us to prove a criterion for compactness of the singular foliation F¯ω, to estimate the number of its minimal components, and to give an upper bound on the rank rkω, in terms of genus