Optimal isoperimetric inequalities for complete proper minimal submanifolds in hyperbolic space

Abstract

Let † be a k-dimensional complete proper minimal submanifold in the Poincaré ball model B n of hyperbolic geometry. If we consider † as a subset of the unit ball B n in Euclidean space, we can measure the Euclidean volumes of the given minimal submanifold † and the ideal boundary @ 1 †, say Vol R . †/ and Vol R .@ 1 †/, respectively. Using this concept, we prove an optimal linear isoperimetric inequality. We also prove that if Vol R .@ 1 †/ Vol R .S k 1 /, then † satisfies the classical isoperimetric inequality. By proving the monotonicity theorem for such †, we further obtain a sharp lower bound for the Euclidean volume Vol R . †/, which is an extension of Fraser-Schoen and Brendle's recent results to hyperbolic space. Moreover we introduce the Möbius volume of † in B n to prove an isoperimetric inequality via the Möbius volume for †.

Linear isoperimetric inequalities for free boundary submanifolds in a geodesic ball

“…In [4], Choe and Gulliver showed this inequality is still true for minimal submanifolds in hyperbolic space. See also [11] for sharp linear isoperimetric inequalities for minimal submanifolds in hyperbolic space.…”

Isoperimetric type inequalities on submanifolds in rotation-symmetric spaces

“…Let Vol R (Σ) denote the volume of Σ ⊂ B 1 with respect to σ Σ , and Vol R (∂Σ) denote the volume of ∂Σ in ∂B 1 . Min and Seo [10] proved the following optimal linear isoperimetric inequality:…”