For higher-order Sobolev inequalities with the sharp constant and only an L^2 remainder, validity is governed by the manifold's scalar curvature and dimension, with a complete answer for k=2.
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Optimal Sobolev inequalities of high order with $L^2$-remainder
For higher-order Sobolev inequalities with the sharp constant and only an L^2 remainder, validity is governed by the manifold's scalar curvature and dimension, with a complete answer for k=2.