Renormalized Area of Catenoids in the Hyperbolic Space

Abstract

We show that the generating curves of non-totally geodesic spherical rotational minimal hypersurfaces (catenoids, for simplicity) of the hyperbolic spaces $\mathbb{H}^{2n+1}$ are $p$-elastic curves for $p=(2n-1)/(2n)$. We employ this variational characterization, together with the Chern--Gauss--Bonnet formulas for locally conformally flat manifolds, to present an explicit expression for the renormalized area of catenoids in terms of hyperelliptic integrals. Further analyzing these special integrals, we show that the renormalized area of catenoids varies continuously from negative infinity to twice the renormalized area of the totally geodesic hypersurfaces $\mathbb{H}^{2n}\subset\mathbb{H}^{2n+1}$. Therefore, we conclude that the renormalized area is not bounded below and that, when $n$ is even, the renormalized area of minimal hypersurfaces in $\mathbb{H}^{2n+1}$ does not have a sign.