Abstract
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.
Highlights
We give an upper bound of the volume of the symmetric difference between a Borel set and a sub-level set of the associated guiding function, in terms of the deficit in Bakry–Ledoux’s Gaussian isoperimetric inequality
Its rigidity and stability are important subjects, for instance in connection with the theory of convergence of spaces
One is based on compactness arguments: we take a sequence of spaces asymptotically satisfying equality in the inequality in question, and apply a rigidity result to its limit space. This method usually provides implicit estimates. Another strategy is an explicit quantitative estimate that we follow in this article
Summary
Geometric and functional inequalities under various curvature bounds are one of the main subjects of comparison geometry and geometric analysis. The use of the reverse Poincare inequality is inspired by [Ma2] where we studied the rigidity problem, and reveals an interesting relation between the isoperimetric inequality and the spectral gap via the guiding function. The reverse Poincare inequality plays an essential role to integrate 1-dimensional estimates on needles into an estimate on M in the proof of the main theorem (precisely, Proposition 7.3 is a key ingredient). This is the starting point of all the estimates in the sequel.
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