This is a very nice article which deserves to be studied carefully by anyone interested in finding solutions to the constraints. In particular, they show how to construct a solution which is far from maximal, and, at the same time, is asymptotically flat. Readers should be aware that the first theorem, Theorem 1.1, covers a much broader range of data than the second theorem, Theorem 1.2. Further, they should be aware that the titles of the theorems ‘Far-from-CMC’ (Theorem 1.1), and ‘Near-CMC’ (Theorem 1.2), especially the second one, are not particularly illuminating.
There are conditions which are surprising. No restriction is placed on Τ2 (other than the AF condition), but we are asked to ensure that KTT2 be small, while, at the same time, the Yamabe constant of the metric be positive! I would not be surprised if the KTT condition was an artefact of the solution technique. This is not to take away from the results derived. They are really admirable and the results significantly increase our understanding of the conformal method of solving the constraints.
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