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~$R^n$.
Citation
Changyu Xia. "Complete manifolds with nonnegative Ricci curvature and almost best Sobolev constant." Illinois J. Math. 45 (4) 1253 - 1259, Winter 2001. https://doi.org/10.1215/ijm/1258138064
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