Certain Gagliardo-Nirenberg and unweighted Yamabe-type constant comparisons force open Riemannian sets to be flat.
CaseA Yamabe-type problem on smooth metric measure spaces, J
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On the rigidity of manifolds with respect to Gagliardo-Nirenberg inequalities
Certain Gagliardo-Nirenberg and unweighted Yamabe-type constant comparisons force open Riemannian sets to be flat.