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 ${\delta }_2(H)\ge (91\mathrm{/}216+\eta )\left(\genfrac{}{}{0px}{}{n}{3}\right)$ contains a Hamilton 2-cycle. This minimum 2-degree condition is asymptotically best possible. Moreover, we give some related results on Hamilton $\mathrm{\ell }$-cycles with $d$-degree for $\mathrm{\ell }\le d\le k-1$ and $1\le \mathrm{\ell }<k\mathrm{/}2$.