Minimum degree thresholds for Hamilton $(k\mathrm{/}2)$-cycles in $k$-uniform hypergraphs

For any even integer $k\ge 6$, integer $d$ such that $k\mathrm{/}2\le d\le k-1$, and sufficiently large $n\in (k\mathrm{/}2)\mathbb{N}$, we find a tight minimum $d$-degree condition that guarantees the existence of a Hamilton $(k\mathrm{/}2)$-cycle in every $k$-uniform hypergraph on $n$ vertices. When $n\in k\mathbb{N}$, the degree condition coincides with the one for the existence of perfect matchings provided by R"odl, Ruci'nski and Szemer'edi (for $d=k-1$) and Treglown and Zhao (for $d\ge k\mathrm{/}2$), and thus our result strengthens theirs in this case.