Tight Hamiltonian Cycles in Uniformly Dense $3$-Graphs

We study minimum degree conditions for tight Hamiltonian cycles in uniformly dense $3$-uniform hypergraphs. We prove that for every $d,\alpha >0$, every sufficiently large $(\rho ,d)$-dense $3$-graph on $n$ vertices with minimum codegree at least $(1\mathrm{/}3+\alpha )n$ contains a tight Hamiltonian cycle. This resolves a problem of Aigner-Horev and Levy in a stronger form, and the constant $1\mathrm{/}3$ is asymptotically best possible. We also show that uniform density does not lower the asymptotic vertex-degree threshold: there are $(\rho ,d)$-dense $3$-graphs with minimum vertex degree $(5\mathrm{/}9-o(1))\left(\genfrac{}{}{0px}{}{n}{2}\right)$ and no tight Hamiltonian cycle. Finally, we construct $(\rho ,2-\sqrt{3})$-dense examples with minimum codegree $(2-\sqrt{3}-o(1))n$ and no tight Hamiltonian cycle, answering negatively a question of Ara{ú}jo, Piga and Schacht.