On weakly keen Heegaard splittings with distance 2

A Heegaard splitting V1∪SV2 is called weakly keen with distance 2 if there are essential separating disks Di⊂Vi such that a component of Vi−Di is homeomorphic to Fi×I and there is an unique geodesic {∂D1,C0,∂D2} in C(S) connecting ∂D1 to ∂D2, where C0 is an essential simple closed curve in S and Fi is a component of ∂−Vi for i=1,2. In this paper, we give a sufficient condition for the weakly keen Heegaard splittings to be keen. At last, we give a sufficient condition for the self-amalgamation of a weakly keen Heegaard splitting to be unstabilized.