The Renormalized Area--Willmore Relation

In [1, Proposition 8.1], Alexakis and Mazzeo showed that the renormalized area of minimal surfaces in the hyperbolic space ${\mathbb{H}}^3$ approaching the ideal boundary ${\mathrm{\partial }}_{\mathrm{\infty }}{\mathbb{H}}^3$ is the negative of the Willmore energy of their Euclidean conformal compactifications. In this paper, we investigate whether there are other geometric surface equations for which these two quantities satisfy a universal proportionality relation. We answer this question in the affirmative by extending the Alexakis–Mazzeo result to a broader class of geometric equations containing, as two distinguished cases, the equations defining generalized singular minimal surfaces in ${\mathbb{R}}^3$ and linear Weingarten surfaces in ${\mathbb{H}}^3$. Moreover, for rotational linear Weingarten surfaces, we express both energies in terms of generalized hyperelliptic integrals and show that their doubles form a family of rotational tori in ${\mathbb{R}}^3$ whose Willmore energies realize all values in $(2{\pi }^2,\mathrm{\infty })$, with a particular subfamily converging to the Clifford torus in ${\mathcal{W}}^{2,p}$ for every $1\le p<\mathrm{\infty }$, as the optimal value $2{\pi }^2$ is approached.