Sharp isoperimetric and Sobolev inequalities in spaces with nonnegative Ricci curvature

Comparison results for solutions of Poisson equations with Robin boundary on complete Riemannian manifolds

“…Let Ω be a bounded domain with smooth boundary on M and Ω ♯ be a geodesic ball on R n (κ) satisfying |Ω| g = α κ |Ω ♯ | g κ . According to Brendle [8] (See also [1,6]) and Lévy-Gromov [20], we have the isoperimetric inequality…”

“…According to isoperimetric inequalities in [8] for κ = 0 (See also [1,6]) and in [20] for κ = 1, we define…”

The isoperimetric problem via direct method in noncompact metric measure spaces with lower Ricci bounds

“…Moreover, the existence of isoperimetric regions has been crucial for the derivation of differential properties of the isoperimetric profile [19,52,21,20,22]. As shown in [17], Theorem 1.1 is a crucial ingredient for proving such properties without assuming existence of isoperimetric regions, as well as, for deriving isoperimetric inequalities and other geometric functional inequalities like the ones contained in [1,18,24].…”

A sharp isoperimetric inequality in metric measure spaces with non-negative Ricci curvature

“…Recently, isoperimetric inequalities in the class of non-compact metric measure spaces with synthetic non-negative Ricci curvature is studied in [1] and [3]. In these isoperimetric inequalities, a key number in the isoperimetric profile is a constant called asymptotic volume ratio of a metric measure space (X, d, m), defined as…”

“…Since the dimension upper bound N appear in the asymptotic volume ratio, the isoperimetric inequalities obtained in [1] and [3] are all dimension-dependent. So it is natural to find a dimension-free version of the isoperimetric inequality for metric measure spaces with non-negative Ricci curvature.…”