In the paper,we give two conditions that the Heegaard splitting admits the disjoint curve property.The main result is that for a genus g(g■2)strongly irreducible Heegaard splitting(C_1,C_2;F),let D_i be an essential disk in C_i,i=1,2,satisfying(1)at least one of ■D_1 and ■D_2 is separating in F and|■D_1∩■D_2|■2g-1;or(2)both ■D_1 and ■D_2 are non-separating in F and|■D_1∩■D_2|■2g-2,then(C_1,C_2;F)has the disjoint curve property.
Let （M; H1, H2; Fo） be a SD-splitting for bordered 3-manifold M. The splitting is reducible （weakly reducible, respectively） if there exist essential disks D1 belong to H1 and D2 belong to H2 such that δD1,δD2 belong to Fo and δD1 =δD2 （δD1 ∩ δD2 =φ, respectively）. A SD-splitting （M; H1, H2; Fo） for bordered 3-manifold M is of inner genus 1 if Fo is a punctured torus. In the present paper, we show that a weakly reducible SD-splitting of inner genus 1 is either reducible or bilongitudional.
