We consider states of the D1-D5 CFT where only the left-moving sector is excited. As we deform away from the orbifold point, some of these states will remain BPS while others can `lift. We compute this lifting for a particular family of D1-D5-P states, at second order in the deformation off the orbifold point. We note that the maximally twisted sector of the CFT is special: the covering surface appearing in the correlator can only be genus one while for other sectors there is always a genus zero contribution. We use the results to argue that fuzzball configurations should be studied for the full class including both extremal and near-extremal states; many extremal configurations may be best seen as special limits of near extremal configurations.
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