Rainbow perfect matchings in 3-partite 3-uniform hypergraphs

Let $m,n,r,s$ be nonnegative integers such that $n\ge m=3r+s$ and $1\le s\le 3$. Let $$\delta (n,r,s)=\{\begin{array}{ll}{n}^2-(n-r{)}^2& \text{if }s=1,\\ n^2-(n-r+1)(n-r-1)& \text{if }s=2,\\ n^2-(n-r)(n-r-1)& \text{if }s=3.\end{array}$$ We show that there exists a constant $n_0>0$ such that if $F_1,\dots ,F_n$ are 3-partite 3-graphs with $n\ge n_0$ vertices in each partition class and minimum vertex degree of $F_i$ is at least $\delta (n,r,s)+1$ for $i\in [n]$ then $\{F_1,\dots ,F_n\}$ admits a rainbow perfect matching. This generalizes a result of Lo and Markström on the vertex degree threshold for the existence of perfect matchings in 3-partite 3-graphs. In this proof, we use a fractional rainbow matching theory obtained by Aharoni et al. to find edge-disjoint fractional perfect matching.