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- W2951497306 abstract "Let (M,g) be a smooth compact Riemannian manifold without boundary of dimension n>=6. We prove that {align*} |u|_{L^{2^*}(M,g)}^2 le K^2int_M{|nabla_g u|^2+c(n)R_gu^2}dv_g +A|u|_{L^{2n/(n+2)}(M,g)}^2, {align*} for all uin H^1(M), where 2^*=2n/(n-2), c(n)=(n-2)/[4(n-1)], R_g is the scalar curvature, $K^{-1}=inf|nabla u|_{L^2(R^n)}|u|_{L^{2n/(n-2)}(R^n)}^{-1}$ and A>0 is a constant depending on (M,g) only. The inequality is {em sharp} in the sense that on any (M,g), $K$ can not be replaced by any smaller number and R_g can not be replaced by any continuous function which is smaller than R_g at some point. If (M,g) is not locally conformally flat, the exponent 2n/(n+2) can not be replaced by any smaller number. If (M,g) is locally conformally flat, a stronger inequality, with 2n/(n+2) replaced by 1, holds in all dimensions n>=3." @default.
- W2951497306 created "2019-06-27" @default.
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- W2951497306 date "2002-01-24" @default.
- W2951497306 modified "2023-09-22" @default.
- W2951497306 title "A sharp Sobolev inequality on Riemannian manifolds" @default.
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