The monotonicity of the ratio of hyperelliptic integrals

In this paper, we study the monotonicity of the ratio of two Abelian integralsI0(h)=∮γ(h)ydx,andI1(h)=∮γ(h)xydx in an interval Σ  , where γ(h)γ(h) is a compact component of hyperelliptic curves with genus 2 as h∈Σh∈Σ. We first give the topological classification of these hyperelliptic curves. Then we show what kinds of compact components in the classification ensure that the ratio of the two Abelian integrals is monotone.