2025/03/17 by Pérez-Ayala, Samuel, Tyrrell, Aaron J.
#Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2503.12703
Let (Mn+1,g+) be an asymptotically hyperbolic manifold. We compute the Cheeger constant of conformally compact asymptotically constant mean curvature submanifolds ι: Yk+1 → (Mn+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 \citeBlitzSamuel2021CFFa, we show that an asymptotically hyperbolic manifold with umbilic boundary is conformally weakly Poincaré--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 Poincaré--Einstein manifold, we identify an extrinsic conformal invariant of ∂ 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 ∂ Y, one which depends on its extrinsic geometry within Y and the other on its extrinsic geometry within ∂ 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 Poincaré--Einstein space, being weakly Poincaré--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.