Complete manifolds with nonnegative Ricci curvature and almost best Sobolev constant

Abstract

We prove that for any given integer $n\geq 2$ and $q\in [1, n)$ there exists a constant $\epsilon= \epsilon(n,q)>0$ such that any $n$-dimensional complete Riemannian manifold with nonnegative Ricci curvature, in which the Sobolev inequality

\[ \left(\int_M|f|^{\frac {nq}{n-q}}\,dv\right)^{\frac{n-q}{nq}}\leq (K(n,q)+\epsilon)\left(\int_M|\nabla f|^q \,dv\right)^{\sfrac{1}{q}}, \,\,\forall f\in C_0^{\infty}(M) \]

holds with $K(n,q)$ the optimal constant of this inequality in the $n$-dimensional Euclidean space $R^n$, is diffeomorphic to~$R^n$.