Holography and Cheeger constant of asymptotically CMC submanifolds

Let $(M^{n+1},g_+)$ be an asymptotically hyperbolic manifold. We compute the Cheeger constant of conformally compact asymptotically constant mean curvature submanifolds $ \iota : Y^{k+1} \to (M^{n+1},g_+)$ with arbitrary codimension. As an application, we provide two classes of examples of $(n+1)$-dimensional asymptotically hyperbolic manifolds with Cheeger constant equal to $n$, whose conformal infinity is of the following types: 1) positive Yamabe invariant, and 2) negative Yamabe invariant. Moreover, in the same spirit as Blitz--Gover--Waldron \cite{BlitzSamuel2021CFFa}, we show that an asymptotically hyperbolic manifold with umbilic boundary is conformally weakly Poincar\'e--Einstein if and only if the third conformal fundamental form of the boundary vanishes. Next, in the space of asymptotically minimal hypersurfaces $Y$ within a Poincar\'e--Einstein manifold, we identify an extrinsic conformal invariant of $\partial Y$ which obstructs the vanishing of the mean curvature of $Y$ to second order. This conformal invariant is a linear combination of two Riemannian hypersurface invariants of $\partial Y,$ one which depends on its extrinsic geometry within $\overline{Y}$ and the other on its extrinsic geometry within $\partial M;$ neither of which are conformal invariants individually. Finally, we show that for asymptotically minimal hypersurfaces with mean curvature vanishing to second order inside of a Poincar\'e--Einstein space, being weakly Poincar\'e--Einstein is equivalent to the boundary of $Y$ having vanishing second and third conformal fundamental forms when viewed as a hypersurface within the conformal infinity.