Sharp Sobolev inequalities on noncompact Riemannian manifolds with bounded Ricci curvature

Given a smooth, complete Riemannian manifold $M$ with bounded Ricci curvature and positive injectivity radius, we derive a sharp Sobolev inequality for the embedding of $W^{1,p}(M)$ into $L^{\frac{np}{n-p}}(M)$, when $1\le p< n$. We will first reduce the inequality to functions having support with small enough volume. In turn, we will show that the inequality for small volumes is implied by a first order uniform asymptotic expansion of the isoperimetric profile for $M$, for small volumes. We will then show that such an expansion follows from a local, uniform Sobolev inequality for functions in $W^{1,1}$, having support with small enough diameter.

💡 Research Summary

The paper establishes a sharp Sobolev inequality on complete non‑compact Riemannian manifolds whose Ricci curvature is bounded and whose injectivity radius is strictly positive. Precisely, for any dimension (n\ge2) and any exponent (1\le p<n) the authors prove that there exist constants (K(n,p)) (the Euclidean optimal Sobolev constant) and a finite (B=B(n,p,K,\operatorname{inj}(M))) such that for every (u\in W^{1,p}(M))
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