On manifolds with non-negative Ricci curvature and Sobolev inequalities

Abstract

— Let M be a complete n-dimensional Riemanian manifold with non-negative Ricci curvature in which one of the Sobolev inequalities (∫ |f |dv )1/p ≤ C(∫ |∇f |qdv)1/q, f ∈ C∞ 0 (M), 1 ≤ q 1. Moreover, for q > 1, the equality in (1) is attained by the functions (λ + |x|q/(q−1))1−(n/q), λ > 0, where |x| is the Euclidean length of the vector x in IR. We are actually interested here in the geometry of those manifolds M for which one of the Sobolev inequalities (1) is satisfied with the best constant C = K(n, q) of IR. The result of this note is the following theorem. Theorem. Let M be a complete n-dimensional Riemannian manifold with nonnegative Ricci curvature. If one of the Sobolev inequalities (1) is satisfied with C = K(n, q), then M is isometric to IR. The particular case q = 1 (p = n/(n − 1)) is of course well-known. In this case indeed, the Sobolev inequality is equivalent to the isoperimetric inequality ( voln(Ω) )(n−1)/n ≤ K(n, 1)voln−1(∂Ω) where ∂Ω is the boundary of a smooth bounded open set Ω in M . If we let V (x0, s) = V (s) be the volume of the geodesic ball B(x0, s) = B(s) with center x0 and radius s in M , we have d ds voln ( B(s) ) = voln−1 ( ∂B(s) ) . Hence, setting Ω = B(s) in the isoperimetric inequality, we get V (s)(n−1)/n ≤ K(n, 1)V ′(s) for all s. Integrating yields V (s) ≥ (nK(n, 1))−nsn, and since K(n, 1) = n−1ω n , for every s, (2) V (s) ≥ V0(s) where V0(s) = ωns is the volume of the Euclidean ball of radius s in IR. IfM has nonnegative Ricci curvature, by Bishop’s comparison theorem (cf. e.g. [Ch]) V (s) ≤ V0(s) for every s, and by (2) and the case of equality,M is isometric to IR. The main interest of the Theorem therefore lies in the case q > 1. As usual, the classical value q = 2 (and p = 2n/(n − 2)) is of particular interest (see below). It should be noticed that known results already imply that the scalar curvature of M is zero in this case (cf. [He], Prop. 4.10). Proof of the Theorem. It is inspired by the technique developed in the recent work [B-L] where a sharp bound on the diameter of a compact Riemannian manifold satisfying a Sobolev inequality is obtained, extending the classical Myers theorem. We thus assume that the Sobolev inequality (1) is satisfied with C = K(n, q) for some q > 1. Recall first that the extremal functions of this inequality in IR are the functions (λ + |x|q)1−(n/q), λ > 0, where q′ = q/(q − 1). Let now x0 be a fixed point in M and let θ > 1. Set f = θ−1d(·, x0) where d is the distance function on M . The idea is then to apply the Sobolev inequality (1), with C = K(n, q), to (λ+ f ′ )1−(n/q), for every λ > 0 to deduce a differential inequality whose solutions may be compared to the extremal Euclidean case. Set, for every λ > 0, F (λ) = 1 n− 1 ∫ 1 (λ+ fq)n−1 dv. Note first that F is well defined and continuously differentiable in λ. Indeed, by Fubini’s theorem, for every λ > 0,