The relative isoperimetric inequality in Cartan-Hadamard 3-manifolds

The double cover relative to a convex domain and the relative isoperimetric inequality

“…Recently, the relative isoperimetric inequality in M 3 was proved in [6]. In this paper we prove the inequality in M 4 .…”

The relative isoperimetric inequality outside convex domains in R n

“…Reasoning as in [4], one can easily see that equality is never attained if C is strictly convex. The analysis of equality in the isoperimetric inequality for a general convex set cannot be treated with the tools used in this paper.…”

“…For n 3 some partial results were known: Kim [11] proved (1) for C = U × R, where U is the epigraph of a C 2 convex function, and Choe [2] proved (1) when ∂D ∩ ∂C is a graph which is symmetric about (n − 1) hyperplanes of R n . More recently, Choe and Ritoré [4] have shown that (1) holds outside convex sets in 3D Cartan-Hadamard manifolds, with equality if and only if D is a flat half ball and := ∂D ∼ ∂C is a hemisphere. The main ingredients of the proof in [4] are the estimate (sup H 2 ) area 2π, and the analysis of the equality case, where H is the mean curvature of ; however, the methods used in [4], which were inspired by the work of Li and Yau [12], are valid only when n = 3.…”

“…More recently, Choe and Ritoré [4] have shown that (1) holds outside convex sets in 3D Cartan-Hadamard manifolds, with equality if and only if D is a flat half ball and := ∂D ∼ ∂C is a hemisphere. The main ingredients of the proof in [4] are the estimate (sup H 2 ) area 2π, and the analysis of the equality case, where H is the mean curvature of ; however, the methods used in [4], which were inspired by the work of Li and Yau [12], are valid only when n = 3.…”

“…When (i), (ii) and (iii) hold it is then known that I m is an absolutely continuous function. For a proof of (i), (ii) and (iii) we refer the reader to [4].…”