Sharp Sobolev-Poincaré inequalities on compact Riemannian manifolds (Q2759061)

Let \((M,g)\) be a smooth compact Riemannian \(n\)-manifold, \(n\geq 4\). If the scalar curvature of \(g\) is negative then the author proves the sharp Sobolev-Poincaré inequality \(\|u\|_{2^*}^2\leq K_n^2\|\nabla u\|_2^2+B(g)\|u\|_1^2\), where \(2^*=2n/(n-2)\), \(\|\cdot\|_p\) denotes the norm in \(L^p(M)\), \(K_n\) is the sharp Euclidean Sobolev constant and \(B(g)\) is the smallest constant for which the inequality holds. He shows that the problem possesses extremal functions and he also studies the case when \(g\) is nonpositive only.