Sharp isoperimetric and Sobolev inequalities in spaces with nonnegative Ricci curvature

On manifolds with non-negative Ricei curvature and Sobolev inequalities

The sharp isoperimetric inequality for non-compact Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth has been obtained in increasing generality with different approaches in a number of contributions culminated by Balogh and Kristály (2023) also covering metric-measure spaces satisfying the nonnegative Ricci curvature condition in the synthetic sense of Lott, Sturm and Villani. In sharp contrast with the compact case of positive Ricci curvature, for a large class of spaces including weighted Riemannian manifolds, no complete characterization of the equality cases is present in the literature. The scope of this paper is to settle this problem by proving, in the same generality as Balogh and Kristály (2023), that the equality in the isoperimetric inequality can be attained only by metric balls. Whenever this happens the space is forced, in a measure theoretic sense, to be a cone. Our result applies to different frameworks yielding as corollaries new rigidity results: it extends the rigidity results of Brendle (2023) for weighted Riemannian manifolds and the rigidity results of Antonelli et al. (2023) for general \mathsf{RCD} spaces. It also applies to the Euclidean setting by proving that optimizers in the anisotropic and weighted isoperimetric inequality for Euclidean cones are necessarily the Wulff shapes.

Minimising hulls, p-capacity and isoperimetric inequality on complete Riemannian manifolds

In this paper, we establish higher-order Sobolev and Rellich-type inequalities on non-compact Riemannian manifolds supporting an isoperimetric inequality. We highlight two notable settings: manifolds with non-negative Ricci curvature and having Euclidean volume growth (supporting Brendle's isoperimetric inequality), and manifolds with non-positive sectional curvature (satisfying the Cartan–Hadamard conjecture or supporting Croke's isoperimetric inequality). Our proofs rely on various symmetrization techniques, and the key ingredient is an iterated Talenti's comparison principle. The non-iterated version is analogous to the main result of Chen and Li [J. Geom. Anal., 2023].

In their seminal work, Cordero-Erausquin, Nazaret and Villani (Adv Math 182(2):307-332, 2004) proved sharp Sobolev inequalities in Euclidean spaces via Optimal Transport, raising the question whether their approach is powerful enough to produce sharp Sobolev inequalities also on Riemannian manifolds. By using L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document}-optimal transport approach, the compact case has been successfully treated by Cavalletti and Mondino (Geom Topol 21:603-645, 2017), even on metric measure spaces verifying the synthetic lower Ricci curvature bound. In the present paper we affirmatively answer the above question for noncompact Riemannian manifolds with non-negative Ricci curvature; namely, by using Optimal Transport theory with quadratic distance cost, sharp Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-Sobolev and Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-logarithmic Sobolev inequalities (both for p>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>1$$\\end{document} and p=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=1$$\\end{document}) are established, where the sharp constants contain the asymptotic volume ratio arising from precise asymptotic properties of the Talentian and Gaussian bubbles, respectively. As a byproduct, we give an alternative, elementary proof to the main result of do Carmo and Xia (Math 140:818-826, 2004) and subsequent results, concerning the quantitative volume non-collapsing estimates on Riemannian manifolds with non-negative Ricci curvature that support Sobolev inequalities.

In this paper we provide new existence results for isoperimetric sets of large volume in Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth. We find sufficient conditions for their existence in terms of the geometry at infinity of the manifold. As a byproduct we show that isoperimetric sets of big volume always exist on manifolds with nonnegative sectional curvature and Euclidean volume growth. Our method combines an asymptotic mass decomposition result for minimizing sequences, a sharp isoperimetric inequality on nonsmooth spaces, and the concavity property of the isoperimetric profile. The latter is new in the generality of noncollapsed manifolds with Ricci curvature bounded below.

This paper presents the proof of several inequalities by using the technique introduced by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First, the author gives a new and simple proof of a lower bound of Berestycki, Nirenberg, and Varadhan concerning the principal eigenvalue of an elliptic operator with bounded measurable coefficients. The rest of the paper is a survey on the proofs of several isoperimetric and Sobolev inequalities using the ABP technique. This includes new proofs of the classical isoperimetric inequality, the Wulff isoperimetric inequality, and the Lions-Pacella isoperimetric inequality in convex cones. For this last inequality, the new proof was recently found by the author, Xavier Ros-Oton, and Joaquim Serra in a work where new Sobolev inequalities with weights came up by studying an open question raised by Haim Brezis.

Some sharp isoperimetric-type inequalities on Riemannian manifolds

Sobolev and isoperimetric inequalities with monomial weights

We are concerned with Lp a priori estimates for the Green function of a regular domain of the Laplacian–Beltrami operator on any 3≤n-dimensional complete noncompact boundary-free Riemannian manifold via rough and sharp isoperimetric-type inequalities. Consequently, we are led to evaluate the critical limit of an induced monotone Green’s functional using the asymptotic behavior of the Lorentz norm deficit of the Green function at infinity, as well as the harmonic radius of a regular domain in the Riemannian manifold with both non-negative Ricci curvature and optimal isoperimetric inequality of Euclidean type.

Analytic inequalities, isoperimetric inequalities and logarithmic Sobolev inequalities

Schneider introduced an inter-dimensional difference body operator on convex bodies, and proved an associated inequality. In the prequel to this work, we showed that this concept can be extended to a rich class of operators from convex geometry and proved the associated isoperimetric inequalities. The role of cosine-like operators, which generate convex bodies in Rn from those in Rn, were replaced by inter-dimensional simplicial operators, which generate convex bodies in Rnm from those in Rn (or vice versa). In this work, we treat the Lp extensions of these operators, and, furthermore, extend the role of the simplex to arbitrary m-dimensional convex bodies containing the origin. We establish mth-order Lp isoperimetric inequalities, including the mth-order versions of the Lp Petty projection inequality, Lp Busemann-Petty centroid inequality, Lp Santaló inequalities, and Lp affine Sobolev inequalities. As an application, we obtain isoperimetric inequalities for the volume of the operator norm of linear functionals (Rn,‖⋅‖E)→(Rm,‖⋅‖F).

We introduce some methods to study the isoperimetric problem in 3-dimensional Riemannian manifolds and we show that in the positive curva- ture case we can control the topology of the isoperimetric regions. We consider the case of the projective space, which was first solved by Ritorea nd Ros, and we apply it to the Willmore problem. The isoperimetric problem is a classical topic in geometry but at the same time many basic questions about it remain unsolved. In this paper we first introduce some of the methods used in the study of that problem, although we will not try to be exhaustive at all. We will explain some relatively flexible ideas, like symmetrization or stability, which can be adapted to a certain number of situations. As an example we will study the problem for radial metrics on the 3-sphere. Then we will show that in 3-manifolds with positive Ricci curvature the topol- ogy of the isoperimetric regions can be controled. In particular we will prove that, when the volume of the ambient space is large, any isoperimetric surface must be either an sphere or a torus. As consequence, we will solve the isoperimetric problem in the real projective space or, equivalently, the isoperimetric problem for antipodal invariant regions in the 3-sphere. That result was first obtained by Ritorea nd Ros (35), but here we will give a somewhat different proof. Finally, as application of the above results, we will solve the Willmore conjec- ture for tori in euclidean space which are symmetric with respect to a point. 2. The isoperimetric problem In this paper we will only consider the three dimensional case. Let M be a Riemaniann 3-dimensional manifold with or without boundary and volume V(M ) ∈ )0, ∞). Given a positive number v< V(M ), we want to study the compact surfaces Σ ⊂ M such that

Affine vs. Euclidean isoperimetric inequalities

Various relations between sharp isoperimetric inequalities and volumes of manifolds are studied. In particular, we introduce and estimate sharp isoperimetric constants τ* and γ* corresponding to two types of isoperimetric inequalities. We show that for a complete n-dimensional manifold M with Ricci curvature Ric(M) ≥ n - 1, the volume of M is close to that of S n if and only if τ* is close to n(n - 1)/(2(n + 2)ω 2/n n ) and M is simply connected (for n = 2 or 3), or γ* is close to 1 (for any n ≥ 2).