Isoperimetric inequality for exterior domains on $\mathbb{H}^{n}$

Question

Fix $n\geq 2$ and let $\mathbb{H}^{n}=\mathbb{R}_{+}\times \mathbb{S}^{n-1}$ be the hyperbolic space be defined as a Riemannian manifold equipped with the Riemannian metric $$g=dr^{2}+\sinh^{2}rd\theta^{2},$$ where $dr^{2}$ is the standard metric on $\mathbb{R}_{+}$ and $d\theta^{2}$ is the standard metric on the sphere $\mathbb{S}^{n-1}$. Denote with $\mu$ the Riemannian measure on $\mathbb{H}^{n}$ and with $\sigma$ the Riemannian measure of co-dimension $1$ on hypersurfaces on $\mathbb{H}^{n}$. It is a known fact that $\mathbb{H}^{n}$ admits the Isoperimetric inequality $$\sigma(\partial \Omega)\geq f(\mu(\Omega)),$$ for all precompact open sets $\Omega\subset \mathbb{H}^{n}$ with smooth boundary, where $f$ is defined by $$f(v)=\left\{\begin{array}{cl}v, &v\geq 1\\ v^{\frac{n-1}{n}}, &v\leq 1.\end{array}\right.$$

Let us fix some compact set $K\subset \mathbb{H}^{n}$ and consider $\mathbb{H}^{n}\setminus K$ as a manifold. Does the same Isoperimetric inequality now hold for precompact open sets $\Omega\subset \mathbb{H}^{n}\setminus K$ with smooth boundary and if yes, where can I find a reference for this statement? If this is not known for the hyperbolic space $\mathbb{H}^{n}$, is a similar statement known for other Riemannian manifolds, for example $\mathbb{R}^{n}$?

Thanks in advance for your help!

Answer 1

I might be missing something, but if you require $\Omega$ to be precompact in $\mathbb{H}^n \setminus K$, then $K$ makes no difference for the purpose of determining the measures and you can use the original inequality. The same if you only require pre-compactness in $\mathbb{H}^n$ but include the measure of $\partial \Omega \cap K$.

But if you do neither and only consider bounded $\Omega$ and their relative boundary in $\mathbb{H}^n \setminus K$, then generally there is no isoperimetric inequality: Pick any set $\Omega$ of positive measure and $K := \partial \Omega$, then $$\sigma(\partial \Omega \setminus K) = 0 < f(\mu(\Omega)).$$