Explicit sharp constants in Sobolev inequalities on Riemannian manifolds with nonnegative Ricci curvature

Alexandru Kristály

Published 2020-12-22Version 1

Let $(M,g)$ be a noncompact, complete $n$-dimensional Riemannian manifold with nonnegative Ricci curvature and Euclidean volume growth, i.e., $0<$AVR$(g)\leq 1$, where AVR$(g)$ stands for the asymptotic volume ratio of $(M,g)$. The main purpose of the paper is to prove that the sharp Sobolev constants in various $L^p$-Sobolev inequalities on $(M,g)$ (both for $p<n$ and $p>n$) can be explicitly given by means of the geometric invariant AVR$(g)$. The equality cases are also characterized by stating that nonzero extremal functions occur if and only if AVR$(g)=1$, i.e., $(M,g)$ is isometric to the standard Euclidean space $(\mathbb R^n,g_0).$ The proofs are based on an isoperimetric inequality established by S. Brendle (2020), combined with appropriate symmetrization techniques and optimal volume non-collapsing properties. We also provide examples of Riemannian manifolds with their explicit asymptotic volume ratios, which represent typical geometric settings we are working in.