Hamiltonicity in randomly perturbed hypergraphs

For integers k≥3 and 1≤ℓ≤k−1, we prove that for any α>0, there exist ϵ>0 and C>0 such that for sufficiently large n∈(k−ℓ)N, the union of a k-uniform hypergraph with minimum vertex degree αn^k−1 and a binomial random k-uniform hypergraph G^(k)(n,p) with p≥n^{−(k−ℓ)−ϵ} for ℓ≥2 and p≥Cn^−(k−1) for ℓ=1 on the same vertex set contains a Hamiltonian ℓ-cycle with high probability. Our result is best possible up to the values of ϵ and C and answers a question of Krivelevich, Kwan and Sudakov.