Minimum vertex degree condition for tight Hamiltonian cycles in 3-uniform hypergraphs

We show that every 3-uniform hypergraph with $n$ vertices and minimum vertex degree at least $(5/9+o(1))\binom{n}2$ contains a tight Hamiltonian cycle. Known lower bound constructions show that this degree condition is asymptotically optimal.
Comment: 38 pages, second version addresses changes arising from the referee reports