Continuing our previous work (Cohn, Lam, Lu, Yang, Nonlinear Analysis, 2011), we obtain a class of Trudinger-Moser inequalities on the entire Heisenberg group, which indicate what the best constants are. All the existing proofs of similar inequalities on unbounded domain of the Euclidean space or the Heisenberg group are based on rearrangement argument. In this note, we propose a new approach to solve this problem. Specifically we get the global Trudinger-Moser inequality by gluing local estimates with the help of cut-off functions. Our method still works for similar problems when the Heisenberg group is replaced by the Euclidean space or complete noncompact Riemannian manifolds.

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