We establish a quantitative isoperimetric inequality for weighted Riemannian manifolds with \operatorname{Ric}_{\infty} \ge 1 . Precisely, we give an upper bound of the volume of the symmetric difference between a Borel set and a sub-level (or super-level) set of the associated guiding function (arising from the needle decomposition), in terms of the deficit in Bakry–Ledoux’s Gaussian isoperimetric inequality. This is the first quantitative isoperimetric inequality on noncompact spaces besides Euclidean and Gaussian spaces. Our argument makes use of Klartag’s needle decomposition (also called localization), and is inspired by a recent work of Cavalletti, Maggi and Mondino on compact spaces. Besides the quantitative isoperimetry, a reverse Poincaré inequality for the guiding function that we have as a key step, as well as the way we use it, are of independent interest.

We study, on a weighted Riemannian manifold of Ric N ≥ K > 0 for N < −1, when equality holds in the isoperimetric inequality. Our main theorem asserts that such a manifold is necessarily isometric to the warped product R × cosh( √ K/(1−N )t) Σ n−1 of hyperbolic nature, where Σ n−1 is an (n − 1)-dimensional manifold with lower weighted Ricci curvature bound and R is equipped with a hyperbolic cosine measure. This is a similar phenomenon to the equality condition of Poincaré inequality. Moreover, every isoperimetric minimizer set is isometric to a half-space in an appropriate sense.In a weighted manifold, the Ricci curvature is modified into the weighted Ricci curvature Ric N involving a parameter N which is sometimes called the effective dimension. Recently, the developments on the curvature-dimension condition in the sense of Lott, Sturm and Villani have shed new light on the theory of curvature bounds for weighted manifolds. The curvature-dimension condition CD(K, N) is a synthetic notion of lower Ricci curvature bounds for metric measure spaces. The parameters K and N are usually regarded as "a lower bound of the Ricci curvature" and "an upper bound of the dimension", respectively. The roles of K and N are better understood when we consider a weighted Riemannian manifold (M, g, m):Geometric analysis for the weighted Ricci curvature Ric N (also called the Bakry-Émery-Ricci curvature) including the isoperimetric inequality has been intensively studied by Bakry and his collaborators in the framework of the Γ-calculus (see [BGL] and [BL]). Recently it turned out that there is a rich theory also for N ∈ (−∞, 1], though this range seems strange due to the above interpretation of N as an upper dimension bound. For examples, various Poincaré-type inequalities ([KM]), the curvature-dimension condition ([Oh3, Oh2]), the splitting theorem ([Wy]) were studied for N < 0 or N ≤ 1. In our previous paper [Mai], we studied the rigidity of the Poincaré inequality (spectral gap) under the condition Ric N ≥ K > 0 with N < −1, and showed that the sharp spectral gap is achieved only if the space is isometric to the warped product R × cosh( √ K/(1−N )t) Σ n−1 of hyperbolic nature. In this paper, we continue this study to the rigidity problem of the isoperimetric inequality.The isoperimetric inequality on weighted manifolds satisfying Ric N ≥ K and diam(M) ≤ D was studied in [Mi1] and [Mi2]. The isoperimetric inequality could also be verified in a gentle way called the needle decomposition on Riemannian manifolds developed by Klartag in [Kl]. The idea is to reduce a high dimensional inequality into its one dimensional version on geodesics, which is much easier to verify. Cavalletti and Mondino generalized this method to metric measure spaces satisfying CD(K, N) condition for N ∈ (1, ∞), and also established the rigidity result for the isoprimetic inequality in [CM]. In this paper, we use the needle decomposition method to consider the rigidity of the isoperimetric inequality under the condition Ric N ≥ K > 0 and N < −1. The splitting...

In this paper, we investigate complete Riemannian manifolds satisfying the lower weighted Ricci curvature bound Ric N ≥ K with K > 0 for the negative effective dimension N < 0. We analyze two one-dimensional examples of constant curvature Ric N ≡ K with finite and infinite total volumes. We also discuss when the first non-zero eigenvalue of the Laplacian takes its minimum under the same condition Ric N ≥ K > 0, as a counterpart to the classical Obata rigidity theorem. Our main theorem shows that, if N < −1 and the minimum is attained, then the manifold splits off the real line as a warped product of hyperbolic nature.

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Concerning quantitative isoperimetry for a weighted Riemannian manifold satisfying Ric ∞ ⩾ 1, we give an 𝐿 1 -estimate exhibiting that the push-forward of the reference measure by the guiding function (arising from the needle decomposition) is close to the Gaussian measure. We also show 𝐿 𝑝 -and 𝑊 2 -estimates in the 1-dimensional case.M S C 2 0 2 0 49Q10 (primary), 49Q20, 53C21 (secondary) INTRODUCTIONThis short article is devoted to several further applications of the detailed estimates in [15] to quantitative isoperimetry. In [15], on a weighted Riemannian manifold (𝑀, g, 𝔪) (with 𝔪 = e −Ψ vol g ) satisfying 𝔪(𝑀) = 1 and Ric ∞ ⩾ 1, we investigated the stability of the Bakry-Ledoux isoperimetric inequality [1]:for any Borel set 𝐴 ⊂ 𝑀, where 𝖯(𝐴) is the perimeter of 𝐴, 𝜸(𝑑𝑥) = (2𝜋) −1∕2 e −𝑥 2 ∕2 𝑑𝑥 is the Gaussian measure on ℝ, and  (ℝ,𝜸) is its isoperimetric profile written asIt is known by [16, Theorem 18.7] (see also [14, § 3]) that equality holds in (1.1) for some 𝐴 with 𝜃 = 𝔪(𝐴) ∈ (0, 1) if and only if (𝑀, g, 𝔪) is isometric to the product of (ℝ, | ⋅ |, 𝜸) and a weighted