Optimal isoperimetric inequalities for complete proper minimal submanifolds in hyperbolic space

Abstract

Abstract Let Σ be a k-dimensional complete proper minimal submanifold in the Poincaré ball model Bn of hyperbolic geometry. If we consider Σ as a subset of the unit ball Bn in Euclidean space, we can measure the Euclidean volumes of the given minimal submanifold Σ and the ideal boundary ∂ ∞ Σ $\partial _\infty \Sigma $ , say Vol ℝ ( Σ ) $\operatorname{Vol}_{\mathbb {R}}(\Sigma )$ and Vol ℝ ( ∂ ∞ Σ ) $\operatorname{Vol}_{\mathbb {R}}(\partial _\infty \Sigma )$ , respectively. Using this concept, we prove an optimal linear isoperimetric inequality. We also prove that if Vol ℝ ( ∂ ∞ Σ ) ≥ Vol ℝ ( 𝕊 k - 1 ) $\operatorname{Vol}_{\mathbb {R}}(\partial _\infty \Sigma ) \ge \operatorname{Vol}_{\mathbb {R}}(\mathbb {S}^{k-1})$ , then Σ satisfies the classical isoperimetric inequality. By proving the monotonicity theorem for such Σ, we further obtain a sharp lower bound for the Euclidean volume Vol ℝ ( Σ ) $\operatorname{Vol}_{\mathbb {R}}(\Sigma )$ , which is an extension of Fraser–Schoen and Brendle's recent results to hyperbolic space. Moreover we introduce the Möbius volume of Σ in Bn to prove an isoperimetric inequality via the Möbius volume for Σ.