Isoperimetric inequalities on minimal submanifolds of space forms

For a domain U on a certain k-dimensional minimal submanifold of S ⁿ or H ⁿ, we introduce a "modified volume"M(U) of U and obtain an optimal isoperimetric inequality for U k ^k ω _k M (D) ^k-1 ≤Vol(∂D) ^k, where ω _k is the volume of the unit ball of R ^k . Also, we prove that if D is any domain on a minimal surface in S ₊ ⁿ (or H ⁿ, respectively), then D satisfies an isoperimetric inequality 2π A≤L ²+A² (2π A≤L²-A² respectively). Moreover, we show that if U is a k-dimensional minimal submanifold of H ⁿ, then (k-1) Vol(U)≤Vol(∂U).