Sharp Sobolev-Poincaré inequalities on compact Riemannian manifolds
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Abstract:
Given $(M,g)$ a smooth compact Riemannian $n$-manifold, $n \ge 3$, we return in this article to the study of the sharp Sobolev-Poincaré type inequality \begin{equation*}\Vert u\Vert _{2^\star }^2 \le K_n^2\Vert \nabla u\Vert _2^2 + B\Vert u\Vert _1^2\tag *{(0.1)}\end{equation*} where $2^\star = 2n/(n-2)$ is the critical Sobolev exponent, and $K_n$ is the sharp Euclidean Sobolev constant. Druet, Hebey and Vaugon proved that $(0.1)$ is true if $n = 3$, that $(0.1)$ is true if $n \ge 4$ and the sectional curvature of $g$ is a nonpositive constant, or the Cartan-Hadamard conjecture in dimension $n$ is true and the sectional curvature of $g$ is nonpositive, but that $(0.1)$ is false if $n \ge 4$ and the scalar curvature of $g$ is positive somewhere. When $(0.1)$ is true, we define $B(g)$ as the smallest $B$ in $(0.1)$. The saturated form of $(0.1)$ reads as \begin{equation*}\Vert u\Vert _{2^\star }^2 \le K_n^2\Vert \nabla u\Vert _2^2+B(g)\Vert u\Vert _1^2. \tag *{(0.2)}\end{equation*} We assume in this article that $n \ge 4$, and complete the study by Druet, Hebey and Vaugon of the sharp Sobolev-Poincaré inequality $(0.1)$. We prove that $(0.1)$ is true, and that $(0.2)$ possesses extremal functions when the scalar curvature of $g$ is negative. A fairly complete answer to the question of the validity of $(0.1)$ under the assumption that the scalar curvature is not necessarily negative, but only nonpositive, is also given.References
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Additional Information
- Emmanuel Hebey
- Affiliation: Département de Mathématiques, Site de Saint-Martin, Université de Cergy-Pontoise, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France
- Email: Emmanuel.Hebey@math.u-cergy.fr
- Received by editor(s): November 18, 2000
- Published electronically: November 1, 2001
- © Copyright 2001 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 354 (2002), 1193-1213
- MSC (2000): Primary 58E35
- DOI: https://doi.org/10.1090/S0002-9947-01-02913-0
- MathSciNet review: 1867378