Minimum vertex degree threshold for loose Hamilton cycles in 3-uniform hypergraphs

We show that for sufficiently large $n$, every 3-uniform hypergraph on $n$ vertices with minimum vertex degree at least $\binom{n-1}2 - \binom{\lfloor\frac34 n\rfloor}2 + c$,where$c=2$if$n\in 4$\mathbb{N}$$and$c=1$if$n\in 2$\mathbb{N}$\setminus 4$\mathbb{N}$$, contains a loose Hamilton cycle. This degree condition is best possible and improves on the work of Bu\ss, H`an and Schacht who proved the corresponding asymptotical result.