Mobius Energy Research Articles

Decomposition of generalized O’Hara’s energies

O’Hara introduced several functionals as knot energies. One of them is the Mobius energy. We know its Mobius invariance from Doyle-Schramm’s cosine formula. It is also known that the Mobius energy was decomposed into three components keeping the Mobius invariance. The first component of decomposition represents the extent of bending of the curves or knots, while the second one indicates the extent of twisting. The third one is an absolute constant. In this paper, we show a similar decomposition for generalized O’Hara energies. We also extend the cosine formula for the Mobius energy to generalized O’Hara energies. It gives us a condition for which the right circle minimizes the energy under the length-constraint. Furthermore, it shows us how far the energy is from the Mobius invariant property. Using decomposition, the first and second variation formulae are derived.

Upper and Lower Bounds and Modulus of Continuity of Decomposed Möbius Energies

The Mobius energy is one of the knot energies, and is named after its Mobius invariant property. It is known to have several different expressions. One is in terms of the cosine of conformal angle, and is called the cosine formula. Another is the decomposition into Mobius invariant parts, called the decomposed Mobius energies. Hence the cosine formula is the sum of the decomposed energies. This raises a question. Can each of the decomposed energies be estimated by the cosine formula? Here we give an affirmative answer: the upper and lower bounds, and modulus of continuity of decomposed parts can be evaluated in terms of the cosine formula. In addition, we provide estimates of the difference in decomposed energies between the two curves in terms of Mobius invariant quantities.

The $$ L^2 $$ L 2 -gradient of decomposed Möbius energies

The Mobius energy is an example of a knot energy, so-called since it is invariant under Mobius transformations. It has been shown that it is defined on the fractional Sobolev space \( H^{ \frac{3}{2} } \cap W^{1, \infty } \), where it can be decomposed into three parts, each of which retains the Mobius invariance. These results hold not only for knots in three-dimensional Euclidean space but also for closed curves embedded in n -dimensional Euclidean space. It has already been obtained that the variational formulae of the decomposed energies for both curves and their variations are in the same fractional Sobolev space. A formal integration by parts implies that the formulae extend to linear forms for variationals in \( L^2 \) if the curve is in \( H^3 \). Such a linear form is called the \( L^2 \)-gradient. In this paper, we confirm this expectation, and give an explicit expression of the \( L^2 \)-gradient for the decomposed Mobius energies including all lower-order terms.

A decomposition theorem of the Möbius energy II: variational formulae and estimates

It was shown in a preceding paper that the Mobius energy can be decomposed into three parts and the Mobius invariance of each part was investigated. In this paper, we provide analytic application of our decomposition. Variational formulae of the Mobius energy have already been obtained by several mathematicians through quite involved calculations. Explicit expressions and estimates of variational formulae can be derived relatively easily by utilizing our decomposition.

Applications of Almgren-Pitts Min-max theory

We survey recent applications of Almgren-Pitts Min-max theory found by the authors. Namely, the proof of the Willmore conjecture, the proof of the Freedman–He–Wang conjecture on the Mobius energy of links (jointly with Ian Agol), and the proof that manifolds with positive Ricci curvature metrics admit an infinite number of minimal hypersurfaces.

The gradient flow of the Möbius energy near local minimizers

In this article we show that for initial data close to local minimizers of the Mobius energy the gradient flow exists for all time and converges smoothly to a local minimizer after suitable reparametrizations. To prove this, we show that the heat flow of the Mobius energy possesses a quasilinear structure which allows us to derive new short-time existence results for this evolution equation and a Łojasiewicz-Simon gradient inequality for the Mobius energy.

On a flow to untangle elastic knots

In this paper we present a method to untangle smooth knots by a gradient flow for a suitable energy. We show that the flow of smooth initial knots remains smooth for all time and approaches asymptotically an “optimal embedding” in its isotopy type. The method is to set up a gradient flow for the total energy of knots, which consists of bending energy and the Mobius energy of knots.

Mobius Energy of Graphs

In the present paper we introduce Mobius energy for the embedded graphs and formulate its main properties. This energy is invariant under the action of the group generated by all inversions in three-dimensional real space. We study critical configurations for the angles at vertices of degree less than five, and discuss the techniques of construction of symmetric toric embedded graphs with critical values of Mobius energy.

The Euler-Lagrange equation and heat flow for the Möbius energy

We prove the following results: 1 A unique smooth solution exists for a short time for the heat equation associated with the Mobius energy of loops in a euclidean space, starting with any simple smooth loop. 2 A critical loop of the energy is smooth if it has cube-integrable curvature. Combining this with an earlier result of M. Freedman, Z. Wang, and the author, we show that any local minimizer of the energy must be smooth. 3 Circles are the only two-dimensional critical loops with cube-integrable curvature. The technique also applies to a family of other knot energies. Similar problems are open for energies of surfaces or, more generally, for embedded submanifolds in a fixed Riemannian manifold. © 2000 John Wiley & Sons, Inc.

Mobius Energy of Knots and Unknots

Mobius Energy of Knots and Unknots

Torus Knots Extremizing the Möbius Energy

Using the principle of symmetric criticality [Palais 1979], we construct torus knots and links that extremize the Möbiusinvariant energy introduced by O'Hara [1991) and Freedman, He and Wang [1993]. The critical energies are explicitly computable using the calculus of residues, a result obtained in collaboration with Gil Stengle. Experiments with a discretized version of the Mobius energy—applicable to the study of arbitrary knots and links—are also described, and confirm the results of the analytic calculations.