Sharp isoperimetric and Sobolev inequalities in spaces with nonnegative Ricci curvature

Anisotropic symmetrization and Sobolev inequalities on Finsler manifolds with nonnegative Ricci curvature

“…where P F (∂Ω) stands for the anisotropic perimeter of ∂Ω, defined as P F (∂Ω) = ∂Ω dσ F , where dσ F stands for the (n − 1)-dimensional Lebesgue measure induced by dv F . Beside the Riemannian setting, see Brendle [6] and Balogh and Kristály [4], one can characterize the equality in (3.1) in the case of the simplest non-Riemannian Finsler structures, namely on Minkowski spaces. To be precise, let (R n , H) be a Finsler manifold endowed with the Lebesgue measure dv H , such that H : R n → [0, ∞) is a smooth, absolutely homogeneous norm.…”

“…Very recently, Balogh and Kristály [4] proved sharp L p -Sobolev inequalities on n-dimensional Riemannian manifolds having nonnegative Ricci curvature and Euclidean volume growth, whenever 1 < p < n. They used symmetrization arguments and a sharp isoperimetric inequality recently proved by Brendle [6], and alternatively, by themselves [4]. We notice that the sharp isoperimetric inequality in [4] is valid even for generic CD(0, N ) spaces, thus in particular, for reversible Finsler manifolds with nonnegative n-Ricci curvature (for short, Ric n ≥ 0, see §2).…”

“…Based on the sharp isoperimetric inequality in [4], the main purpose of the present paper is to elaborate a suitable symmetrization argument on Finsler manifolds with Ric n ≥ 0 and to establish (possibly sharp) functional inequalities; in particular, we aim to handle the complementary case p > n w.r.t. [4] by establishing sharp Morrey-Sobolev inequalities on such Finsler structures.…”

“…Based on the sharp isoperimetric inequality in [4], the main purpose of the present paper is to elaborate a suitable symmetrization argument on Finsler manifolds with Ric n ≥ 0 and to establish (possibly sharp) functional inequalities; in particular, we aim to handle the complementary case p > n w.r.t. [4] by establishing sharp Morrey-Sobolev inequalities on such Finsler structures. In order to give a flavor of our results that follow from the symmetrization argument, let (M, F ) be a noncompact, complete n-dimensional reversible Finsler manifold with Ric n ≥ 0, endowed with the canonical volume form dv F and the induced (symmetric) Finsler metric d F : M ×M → R, and let F * be the polar transform of F ; for these notions, see §2.…”

“…The key result is a Pólya-Szegő inequality on Finsler manifolds with Ric n ≥ 0, involving the asymptotic volume ratio AVR F , see Theorem 3.1; its proof is based on a symmetrization argument 'from manifolds to normed spaces' in the spirit of Aubin [2], combined with the aforementioned sharp isoperimetric inequality from [4]. One of the main consequences of the latter symmetrization is the following sharp Morrey-Sobolev inequality (with support-bound):…”