Sufficient Conditions for Heegaard Splittings with Disjoint Curve Property

Abstract:

      In the paper, we give two conditions that the Heegaard splitting admits the disjoint curve property. The main result is that for a genus $g~(g\geq 2)$ 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 $\partial D_1$ and $\partial D_2$ is separating in $F$ and $|\partial D_1 \cap \partial D_2 |\leq 2g-1$; or (2) both $\partial D_1$ and $\partial D_2 $ are non-separating in $F$ and $|\partial D_1 \cap \partial D_2 |\leq 2g-2 $, then $(C_1,C_2;F)$ has the disjoint curve property.