Minimum degree conditions for Hamilton $l$-cycles in $ k $-uniform hypergraphs

We show that for $\eta>0$ and sufficiently large $n$, every 5-graph on $n$ vertices with $\delta_{2}(H)\ge (91/216+\eta)\binom{n}{3}$ contains a Hamilton 2-cycle. This minimum 2-degree condition is asymptotically best possible. Moreover, we give some related results on Hamilton $\ell$-cycles with $d$-degree for $\ell\le d \le k-1$ and $1\le \ell < k/2$.