Sharp Sobolev inequalities involving scalar curvature

YanYan Li

Rutgers University

Tonia Ricciardi

Università di Napoli Federico II

Abstract

Sharp Sobolev inequalities are of interest in Geometry and Physics. On Riemannian manifolds, the “interaction” between the sharp Sobolev inequality and the metric structure has attracted much attention in recent years, see [1], [2] and references therein. We are interested in the case of compact manifolds without boundary. Our main result states that, in high dimensions, the “first order term” involves scalar curvature. Namely, let (M, g) be a smooth compact Riemannian manifold without boundary of dimension n ≥ 6. We prove that

∥u∥_{L^{2^{∗}} (M,g)}^{2} ≤ K^{2} ∫_{M} {|∇_{g}u|^{2} + c(n)R_{g}u^{2}} dv_{g} + A∥u∥_{L^{2n/(n+2)}(M,g)}^{2},

for all u ∈ H^{1}(M), where 2^{∗} = 2n/(n − 2), c(n) = (n − 2)/[4(n − 1)], R_{g} is the scalar curvature, K^{−1} = inf ∥∇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 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. The proofs are based on a blow-up analysis of approximate minimizers and on conformal invariance properties.

key words: sharp Sobolev inequality, critical exponent, Yamabe problem MSC 2000 subject classification: 35J60, 58E35

References

[1] T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer Monographs in Math., Springer-Verlag, New York, Berlin, 1998.

[2] E. Hebey, Sobolev spaces on Riemannian manifolds, Lecture Notes in Math. 1635, Springer-Verlag, New York, Berlin, 1996.