A sharp Sobolev inequality on Riemannian manifolds
| dc.creator | Li, YanYan | |
| dc.creator | Ricciardi, Tonia | |
| dc.date | 2002-01-24 | |
| dc.date.accessioned | 2026-07-07T04:46:05Z | |
| dc.date.available | 2026-07-07T04:46:05Z | |
| dc.description | 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^2\int_M\{|\nabla_g u|^2+c(n)R_gu^2\}dv_g +A\|u\|_{L^{2n/(n+2)}(M,g)}^2, {align*} for all u\in 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. | |
| dc.description | 35 pages | |
| dc.identifier | https://arxiv.org/abs/math/0201232 | |
| dc.identifier | http://arxiv.org/abs/math/0201232 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/63192 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | Differential Geometry | |
| dc.subject | 35J60; 58E35 | |
| dc.title | A sharp Sobolev inequality on Riemannian manifolds | |
| dc.type | text |