  For links with vanishing pairwise linking numbers, the link components bound
pairwise disjoint surfaces in $B^{4}$. In this paper, we describe the set of
genera of such surfaces in terms of the $h$-function, which is a link invariant
from Heegaard Floer homology. In particular, we use the $h$-function to give
lower bounds for the 4-genus of the link. For $L$-space links, the $h$-function
is explicitly determined by Alexander polynomials of the link and sublinks. We
show some $L$-space links where the lower bounds are sharp, and also describe
all possible genera of disjoint surfaces bounded by such links.
