Near perfect matchings in uniform hypergraphs

In this paper, we study degree conditions for the existence of large matchings in uniform hypergraphs. We prove that for integers $k,l,n$ with $k\ge 3$,$k/2<l<k$,and$n$large,if$H$isa$k$-uniformhypergraphon$n$verticesand$\delta_{l}(H)>{n-l\choose k-l}-{(n-l)-(\lceil n/k \rceil-2)\choose 2}$,then$H$ has a matching covering all but a constant number of vertices. When $l=k-2$ and $k\ge 5$, such a matching is near perfect and our bound on ${\delta }_l(H)$ is best possible. When $k=3$, with the help of an absorbing lemma of H'{a}n, Person, and Schacht, our proof also implies that $H$ has a perfect matching, a result proved by K" uhn, Osthus, and Treglown and, independently, of Kahn.