On the problem of prescribing weighted scalar curvature and the weighted Yamabe flow

Abstract The weighted Yamabe problem introduced by Case is the generalization of Gagliardo-Nirenberg inequalities to smooth metric measure spaces. More precisely, given a space ( M , g e − ϕ d V m ) \left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) , consists on finding another conformal such that its scalar curvature equal λ + μ ∕ \lambda +\mu {e}^{-\phi /m} for some constants \mu and satisfying certain condition. In this article, we consider prescribing curvature. We first prove uniqueness nonuniqueness results then existence result about also estimate nonzero eigenvalue Laplacian . On other hand, version Schwarz lemma All these are achieved using geometric flows related flow. backward Finally, solitons, which self-similar solutions