The Renormalized Area--Willmore Relation
Abstract
In [1, Proposition 8.1], Alexakis and Mazzeo showed that the renormalized area of minimal surfaces in the hyperbolic space ℍ3 approaching the ideal boundary ∂∞ℍ3 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 ℝ3 and linear Weingarten surfaces in ℍ3. 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 ℝ3 whose Willmore energies realize all values in (2π2,∞), with a particular subfamily converging to the Clifford torus in 𝒲2,p for every 1≤ p<∞, as the optimal value 2π2 is approached.
Related
- Comparison between renormalized volume and area in the Weil--Petersson topology
- Area-constrained Willmore surfaces in Lorentzian manifolds: curvature inequalities and the Hawking quasi-local energy
- The Willmore Energy Landscape of Spheres and Avoidable Singularities of the Willmore Flow
- Willmore energy and minimal fillings
- Eventual Monotonicity of the Willmore Energy under Mean Curvature Flow
BibTeX
@misc{pampano2026renormalized,
title = {{The Renormalized Area--Willmore Relation}},
author = {Alvaro Pampano and Aaron J. Tyrrell},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13661},
eprint = {2610.13661},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13661},
}
RIS
TY - UNPB
TI - The Renormalized Area--Willmore Relation
AU - Alvaro Pampano
AU - Aaron J. Tyrrell
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13661
UR - https://arxiv.org/abs/2610.13661
AB - In [1, Proposition 8.1], Alexakis and Mazzeo showed that the renormalized area of minimal surfaces in the hyperbolic space ℍ^3 approaching the ideal boundary ∂_∞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 ℝ^3 and linear Weingarten surfaces in 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 ℝ^3 whose Willmore energies realize all values in (2π^2,∞), with a particular subfamily converging to the Clifford torus in 𝒲^2,p for every 1≤ p<∞, as the optimal value 2π^2 is approached.
ER -