[Paper Review] Möbius invariant energies and average linking with circles

Jun O’Hara, Gil Solanes|arXiv (Cornell University)|Oct 19, 2010

Mathematics and Applications15 references4 citations

TL;DR

This paper introduces a Möbius-invariant energy for planar domains and space curves via renormalized integrals of geometric potentials, establishing a link to Gauss-Bonnet theorems in hyperbolic space. The key contribution is a closed-form, Möbius-invariant expression for the energy of a space curve involving the angle between tangent planes and linking with circles, proven invariant under Möbius transformations through renormalization and geometric analysis.

ABSTRACT

We define and study a Möbius invariant energy associated to planar domains, as well its generalization to space curves. This generalization is a Möbius version of Banchoff-Pohl's notion of area enclosed by a space curve. A relation with Gauss-Bonnet theorems for complete surfaces in hyperbolic space is also described.

Motivation & Objective

• To define a Möbius-invariant energy for planar domains with smooth boundary, overcoming divergence issues in Auckly-Sadun's renormalized potential.
• To generalize the energy to space curves by interpreting it as a renormalized measure of linked circles, extending Banchoff-Pohl's area concept.
• To establish a Gauss-Bonnet-type formula in hyperbolic 3- and 4-space connecting the energy to curvature and linking numbers.
• To prove Möbius invariance of the energy using geometric renormalization and transformation properties under inversion.

Proposed method

• Define a renormalized potential $ V(w,\Omega) $ for planar domains by subtracting a $ \pi/\varepsilon^2 $ counterterm to cancel divergence near the boundary.
• Introduce the energy $ E(\Omega) $ via a double renormalization: integrating $ V(w,\Omega) $ over $ \Omega_\delta $ and adding a boundary correction term $ \frac{\pi}{4\delta}L(K) $.
• Express the energy $ E(K) $ of a planar curve $ K $ using a contour integral: $ E(K) = -\frac{1}{2}\int_{K\times K} \sin\theta_p \sin\theta_q \frac{dp\,dq}{|q-p|^2} $, eliminating limits.
• Generalize the energy to space curves as $ E(K) = \lim_{\varepsilon\to 0}\left(\frac{3\pi L(K)}{8\varepsilon} - \frac{3}{16\pi}\int_{\mathcal{S}_\varepsilon(1,3)} \lambda^2(\gamma,K)\,d\gamma\right) $, where $ \lambda $ is linking number with circles.
• Prove Möbius invariance by showing the energy remains unchanged under inversion, using asymptotic expansions and geometric identities involving curvature and normal vectors.
• Derive a Gauss-Bonnet formula in $ \mathbb{H}^3 $ and $ \mathbb{H}^4 $, linking the energy to extrinsic curvature, geodesic intersections, and algebraic linking numbers.

Experimental results

Research questions

• RQ1 How can a Möbius-invariant energy be defined for planar domains with non-empty boundary, despite divergence in the Auckly-Sadun potential?
• RQ2 What is the geometric interpretation of the generalized energy for space curves in terms of linked circles?
• RQ3 How does the energy relate to curvature and topology in hyperbolic 3- and 4-manifolds?
• RQ4 Can the energy be expressed without renormalization limits, and is it invariant under Möbius transformations?

Key findings

• The energy $ E(K) $ for a planar curve admits a closed-form expression: $ E(K) = -\frac{1}{2}\int_{K\times K} \sin\theta_p \sin\theta_q \frac{dp\,dq}{|q-p|^2} $, independent of limits.
• For a space curve, the energy is given by $ E(K) = -\frac{1}{2}\int_{K\times K} \cos\tau \sin\theta_p \sin\theta_q \frac{dp\,dq}{|q-p|^2} $, where $ \tau $ is the angle between tangent planes at $ p $ and $ q $.
• The energy $ E(K) $ is Möbius invariant, proven via transformation properties of curvature and normal vectors under inversion.
• A Gauss-Bonnet formula in $ \mathbb{H}^3 $ relates the integral of extrinsic curvature to $ 2\pi\chi(S) $, the linking measure, and $ E(K) $, with explicit correction terms.
• The energy of a planar domain satisfies $ E(\Omega) = E(K) + \frac{\pi^2}{4}\chi(\Omega) $, linking domain and boundary energies.
• The energy for space curves reduces to Banchoff-Pohl area when the power in the denominator is 1, and to length when the power is 0, showing consistency with known invariants.