Cosine formula for generalized O'Hara's energies

TL;DR

This paper extends the cosine formula to generalized O'Hara energies, providing conditions for energy minimization and insights into their deviation from M"obius invariance.

Contribution

It introduces a generalized cosine formula for O'Hara energies and analyzes their minimization and invariance properties.

Findings

Derived a cosine formula for generalized O'Hara energies

Identified conditions for circle minimizers under length constraints

Quantified deviation from M"obius invariance

Abstract

In this short article, we extend the cosine formula for the M\"{o}bius energy to generalized O'Hara energies. The newly derived formula 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 M\"{o}bius invariant property.

Equations

$$\mathcal{E}_{(\alpha,p)}(\mbox{\boldmath$f$})=\iint_{(\mathbb{R}/\mathcal{L}\mathbb{Z})^{2}}\left(\frac{1}{\|\mbox{\boldmath$f$}(s_{1})-\mbox{\boldmath$f$}(s_{2})\|_{\mathbb{R}^{3}}^{\alpha}}-\frac{1}{\mathscr{D}(\mbox{\boldmath$f$}(s_{1}),\mbox{\boldmath$f$}(s_{2}))^{\alpha}}\right)^{p}ds_{1}ds_{2}$$

$$\mathcal{E}_{(2,1)}(\mbox{\boldmath$f$})=\iint_{(\mathbb{R}/\mathcal{L}\mathbb{Z})^{2}}\frac{1-\cos\varphi(s_{1},s_{2})}{\|\mbox{\boldmath$f$}(s_{1})-\mbox{\boldmath$f$}(s_{2})\|_{\mathbb{R}^{n}}^{2}}\,ds_{1}ds_{2}+4,$$

$$\displaystyle\mathcal{E}_{(2,1)}(\mbox{\boldmath$f$})=$$

$$\displaystyle\mathcal{E}_{(2,1),1}(\mbox{\boldmath$f$})=$$

$$\displaystyle\mathcal{E}_{(2,1),2}(\mbox{\boldmath$f$})=$$

$$\mathcal{E}_{\Phi}(\mbox{\boldmath$f$})=\iint_{(\mathbb{R}/\mathcal{L}\mathbb{Z})^{2}}\left(\frac{1}{\Phi(\|\Delta\mbox{\boldmath$f$}\|)}-\frac{1}{\Phi(\mathscr{D}(\mbox{\boldmath$f$}(s_{1}),\mbox{\boldmath$f$}(s_{2})))}\right)ds_{1}ds_{2},$$

$$\displaystyle\mathcal{E}_{\Phi}(\mbox{\boldmath$f$})=$$

$$\displaystyle\mathcal{E}_{\Phi,1}(\mbox{\boldmath$f$})=$$

$$\displaystyle\mathcal{E}_{\Phi,2}(\mbox{\boldmath$f$})=$$

$$\displaystyle\qquad\times\left\langle\mbox{\boldmath$\tau$}(s_{1})\wedge\frac{\Delta\mbox{\boldmath$f$}}{\|\Delta\mbox{\boldmath$f$}\|},\mbox{\boldmath$\tau$}(s_{2})\wedge\frac{\Delta\mbox{\boldmath$f$}}{\|\Delta\mbox{\boldmath$f$}\|}\right\rangle ds_{1}ds_{2},$$

$$\Lambda (t)=$$

$$\mathcal{E}_{\Phi}(\mbox{\boldmath$f$})=\iint_{(\mathbb{R}/\mathcal{L}\mathbb{Z})^{2}}\frac{1-\cos\psi(s_{1},s_{2})}{\Phi(\|\Delta\mbox{\boldmath$f$}\|)}\,ds_{1}ds_{2}+2\mathcal{L}\int_{\frac{\mathcal{L}}{2}}^{\infty}\frac{dt}{\Phi(t)}$$

$$W_{\Phi}=\left\{\mbox{\boldmath$u$}\in W^{1,2}(\mathbb{R}/\mathcal{L}\mathbb{Z})\,\left|\,\iint_{(\mathbb{R}/\mathcal{L}\mathbb{Z})^{2}}\frac{\|\Delta\mbox{\boldmath$u$}^{\prime}\|_{\mathbb{R}^{n}}^{2}}{\Phi(\mathrm{dist}_{\mathbb{R}/\mathcal{L}\mathbb{Z}}(s_{1},s_{2}))}\,ds_{1}ds_{2}<\infty\right.\right\}.$$

$$\mbox{\boldmath$f$}\in W_{\Phi}\cap W^{1,\infty}(\mathbb{R}/\mathcal{L}\mathbb{Z}),\quad\mbox{\boldmath$f$}\mbox{ is bi-Lipschitz}.$$

$$\Lambda (x)=-\frac{1}{x}{\int }_x^{\infty }\frac{dt}{\Phi (t)}.$$

$$\displaystyle\lim_{\varepsilon\to+0}\varepsilon\int_{\mathbb{R}/\mathcal{L}\mathbb{Z}}\left(\Lambda(\|\mbox{\boldmath$f$}(s_{1})-\mbox{\boldmath$f$}(s_{1}+\varepsilon)\|_{\mathbb{R}^{n}})-\Lambda(\varepsilon)\right)ds_{1}=0,$$

$$\displaystyle\lim_{\varepsilon\to+0}\int_{\mathbb{R}/\mathcal{L}\mathbb{Z}}\Lambda(\|\mbox{\boldmath$f$}(s_{1})-\mbox{\boldmath$f$}(s_{1}+\varepsilon)\|_{\mathbb{R}^{n}})\int_{s_{1}}^{s_{1}+\varepsilon}\|\mbox{\boldmath$f$}^{\prime}(s_{1})-\mbox{\boldmath$f$}^{\prime}(s_{2})\|_{\mathbb{R}^{n}}^{2}ds_{2}ds_{1}=0.$$

$${\Theta }_{\Phi }(t)=\frac{1}{2}\left(1+\Phi (t)\Lambda (t)\right)$$

$$\displaystyle\mathcal{E}_{\Phi}(\mbox{\boldmath$f$})=$$

$$\displaystyle\qquad\qquad\qquad\left.+\,\Theta_{\Phi}(\|\Delta\mbox{\boldmath$f$}\|)(1-\cos\psi(s_{1},s_{2}))\right\}\frac{ds_{1}ds_{2}}{\Phi(\|\Delta\mbox{\boldmath$f$}\|)}$$

$$+2\mathcal{L}{\int }_{\frac{\mathcal{L}}{2}}^{\infty }\frac{dx}{\Phi (x)}$$

$$\varphi_{{}_{\Phi}}(s_{1},s_{2})=\arccos\left\{\left(1-\Theta_{\Phi}(\|\Delta\mbox{\boldmath$f$}\|)\right)\cos\varphi(s_{1},s_{2})+\Theta_{\Phi}(\|\Delta\mbox{\boldmath$f$}\|)\cos\psi(s_{1},s_{2})\right\}$$

$$\mathcal{E}_{\Phi}(\mbox{\boldmath$f$})=\iint_{(\mathbb{R}/\mathcal{L}\mathbb{Z})^{2}}\frac{1-\cos\varphi_{{}_{\Phi}}(s_{1},s_{2})}{\Phi(\|\Delta\mbox{\boldmath$f$}\|)}\,ds_{1}ds_{2}+2\mathcal{L}\int_{\frac{\mathcal{L}}{2}}^{\infty}\frac{dx}{\Phi(x)}.$$

$$\displaystyle\mathcal{E}_{\Phi,3}(\mbox{\boldmath$f$})=$$

$$\displaystyle\mathcal{E}_{\Phi,4}(\mbox{\boldmath$f$})=$$

$$\mathcal{E}_{\Phi}(\mbox{\boldmath$f$})=\mathcal{E}_{\Phi,3}(\mbox{\boldmath$f$})+(\mathcal{E}_{\Phi,4}(\mbox{\boldmath$f$})-\mathcal{E}_{\Phi,4}(C))+\mathcal{E}_{\Phi}(C)\geqq(\mathcal{E}_{\Phi,4}(\mbox{\boldmath$f$})-\mathcal{E}_{\Phi,4}(C))+\mathcal{E}_{\Phi}(C).$$

$$\displaystyle\mathcal{E}_{t^{\alpha}}(\mbox{\boldmath$f$})=$$

$$=$$

$$\displaystyle\quad+\,\frac{\alpha-2}{2(\alpha-1)}\iint_{(\mathbb{R}/\mathcal{L}\mathbb{Z})^{2}}\frac{1-\cos\psi(s_{1},s_{2})}{\|\Delta\mbox{\boldmath$f$}\|^{\alpha}}\,ds_{1}ds_{2}+\frac{2^{\alpha}}{(\alpha-1)\mathcal{L}^{\alpha-2}}.$$

$$\displaystyle\mathcal{L}^{\alpha-2}\mathcal{E}_{(\alpha,1)}(\mbox{\boldmath$f$})$$

$$\displaystyle\quad=\left\{1-\frac{\alpha-2}{2(\alpha-1)}\right\}\iint_{(\mathbb{R}/\mathcal{L}\mathbb{Z})^{2}}\left(\frac{\mathcal{L}}{\|\Delta\mbox{\boldmath$f$}\|}\right)^{\alpha-2}(1-\cos\varphi(s_{1},s_{2}))\frac{ds_{1}ds_{2}}{\|\Delta\mbox{\boldmath$f$}\|^{2}}$$

$$\displaystyle\quad\qquad+\,\frac{\alpha-2}{2(\alpha-1)}\iint_{(\mathbb{R}/\mathcal{L}\mathbb{Z})^{2}}\left(\frac{\mathcal{L}}{\|\Delta\mbox{\boldmath$f$}\|}\right)^{\alpha-2}\frac{1-\cos\psi(s_{1},s_{2})}{\|\Delta\mbox{\boldmath$f$}\|^{2}}\,ds_{1}ds_{2}+\frac{2^{\alpha}}{\alpha-1}.$$

$$1-\cos\varphi(s_{1},s_{2}),\quad\frac{ds_{1}ds_{2}}{\|\Delta\mbox{\boldmath$f$}\|^{2}},\quad\iint_{(\mathbb{R}/\mathcal{L}\mathbb{Z})^{2}}\frac{1-\cos\psi(s_{1},s_{2})}{\|\Delta\mbox{\boldmath$f$}\|^{2}}\,ds_{1}ds_{2}$$

$$\left(\frac{\mathcal{L}}{\|\Delta\mbox{\boldmath$f$}\|}\right)^{\alpha-2}$$

$$\displaystyle\mathcal{E}_{\Phi}(\mbox{\boldmath$f$})=$$

$$\displaystyle\qquad\qquad\left.+\,\left(\frac{1}{\Phi(\|\Delta\mbox{\boldmath$f$}\|)}-\Lambda(\|\Delta\mbox{\boldmath$f$}\|)\right)\right.$$

$$\displaystyle\quad\qquad\qquad\left.\times\left\{\mbox{\boldmath$\tau$}(s_{1})\cdot\mbox{\boldmath$\tau$}(s_{2})-\left(\mbox{\boldmath$\tau$}(s_{1})\cdot\frac{\Delta\mbox{\boldmath$f$}}{\|\Delta\mbox{\boldmath$f$}\|}\right)\left(\mbox{\boldmath$\tau$}(s_{2})\cdot\frac{\Delta\mbox{\boldmath$f$}}{\|\Delta\mbox{\boldmath$f$}\|}\right)\right\}\right]ds_{1}ds_{2}$$

$$+2\mathcal{L}{\int }_{\frac{\mathcal{L}}{2}}^{\infty }\frac{dx}{\Phi (x)}$$

$$\cos\varphi(s_{1},s_{2})=-\mbox{\boldmath$\tau$}(s_{1})\cdot\mbox{\boldmath$\tau$}(s_{2})+2\left(\mbox{\boldmath$\tau$}(s_{1})\cdot\frac{\Delta\mbox{\boldmath$f$}}{\|\Delta\mbox{\boldmath$f$}\|}\right)\left(\mbox{\boldmath$\tau$}(s_{2})\cdot\frac{\Delta\mbox{\boldmath$f$}}{\|\Delta\mbox{\boldmath$f$}\|}\right).$$

Full text

keywords: O’Hara’s energy, Möbius energy, knot energy, cosine formula

MSC2010: 53A04, 49Q10 58J70,

1 Introduction

In a series of papers [9]–[11], O’Hara proposed the energy functional

[TABLE]

for an $\mathbb{R}^{3}$-valued function $f$ on $\mathbb{R}/\mathcal{L}\mathbb{Z}$ representing a knot parametrized by the arc-length. Among these, the energy $\mathcal{E}_{(2,1)}$ is prominent for the invariance under Möbius transformations. This property had been first identified by Freedman-He-Wang [3], while Doyle and Schramm later gave an alternative proof showing the expression

[TABLE]

where $\varphi$ is the conformal angle. This is called the cosine formula, and was presented in [8]. The conformal angle is defined as follows. Let $C_{12}$ be the circle contacting a knot \mathrm{Im}\mbox{\boldmathf} at \mbox{\boldmathf}(s_{1}) and passing through \mbox{\boldmathf}(s_{2}). We define the circle $C_{21}$ similarly. The angle $\varphi(s_{1},s_{2})$ is that between these two circles at \mbox{\boldmathf}(s_{1}) (and also at \mbox{\boldmathf}(s_{2})). Note that it is determined from \mbox{\boldmathf}(s_{1})-\mbox{\boldmathf}(s_{2}), \mbox{\boldmathf}^{\prime}(s_{1}), and \mbox{\boldmathf}^{\prime}(s_{2}). It is clearly invariant under Möbius transformations. The invariance of \displaystyle{\frac{ds_{1}ds_{2}}{\|\mbox{\boldmathf}(s_{1})-\mbox{\boldmathf}(s_{2})\|_{\mathbb{R}^{3}}^{2}}} under Möbius transformations follows from that of the cross ratio. Hence, we can easily read the Möbius invariance property of $\mathcal{E}_{(2,1)}$ from (1).

In this article, we discuss an analogue for generalized O’Hara energies. The motivation is as follows. In the following, we use the notation $\Delta$ to mean \Delta\mbox{\boldmathu}=\mbox{\boldmathu}(s_{1})-\mbox{\boldmathu}(s_{2}), and $\|\cdot\|=\|\cdot\|_{\mathbb{R}^{n}}$, $\langle\cdot,\cdot\rangle=\langle\cdot,\cdot\rangle_{{\bigwedge\!}^{2}\,\mathbb{R}^{n}}$. According to [2], if \mathcal{E}_{(2,1)}(\mbox{\boldmathf})<\infty, then the unit tangent vector \mbox{\boldmath\tau}=\mbox{\boldmathf}^{\prime} exists almost everywhere. The author, together with Ishizeki [4, 6] showed a decomposition of $\mathcal{E}_{(2,1)}$ for an $\mathbb{R}^{n}$-valued function:

[TABLE]

Each energy $\mathcal{E}_{(2,1),i}$ is also Möbius invariant. The decomposition was extended for the generalized O’Hara energy

[TABLE]

under suitable assumptions on the function $\Phi$ from $\mathbb{R}_{+}=\{x\in\mathbb{R}\,|\,x>0\}$ to itself, as

[TABLE]

see [7]. Since $\mathcal{E}_{\Phi}$ does not generally have the Möbius invariant property, $\mathcal{E}_{\Phi,i}$ also does not. We, however, can derive the first and second variational formulae from (3) systematically as in [5]. The part “$+\,4$” is common for both (1) and (2). Then (3) suggests that the energy $\mathcal{E}_{\Phi}$ also has the expression

[TABLE]

for some angle $\psi$ determined from \Delta\mbox{\boldmathf}, \mbox{\boldmath\tau}(s_{1}), and \mbox{\boldmath\tau}(s_{2}). The answer is affirmative, and the formula indicates how far the energy is from the Möbius invariant. In next section, we will give the formula and discuss its consequences. The proof of the formula will be given in the last section.

2 Cosine formula

In our previous paper [7] we proved (3) under the assumption that

• (A.1)

$\Phi$ is monotonically increasing.

• (A.2)

$\displaystyle{\int_{x}^{\infty}\frac{dt}{\Phi(t)}<\infty}$ for $x>0$.

• (A.3)

The function space $W_{\Phi}$ is defined by

[TABLE]

If \mathcal{E}_{\Phi}(\mbox{\boldmathf})<\infty, then

[TABLE]

• (A.4)

We set

[TABLE]

If \mbox{\boldmathf}\in W_{\Phi}\cap W^{1,\infty}(\mathbb{R}/\mathcal{L}\mathbb{Z}) is bi-Lipschitz and \|\mbox{\boldmathf}^{\prime}\|_{\mathbb{R}^{n}}\equiv 1 almost everywhere, then it holds that

[TABLE]

• (A.5)

• (a)

For any $\lambda\in(0,1)$ and any $x\in\left(0,\frac{\mathcal{L}}{2}\right]$, there is a constant $C(\lambda,\mathcal{L})>0$ such that $\Phi(\lambda x)\geqq C(\lambda,\mathcal{L})\Phi(x)$,

• (b)

$\displaystyle{\inf_{x\in\left(0,\frac{\mathcal{L}}{2}\right]}\left(\frac{1}{\Phi(x)}+\Lambda(x)\right)\geqq 0}$.

In the case of O’Hara $\Phi(t)=t^{\alpha}$, the above assumptions hold if $\alpha\in[2,3)$ where the functional $\mathcal{E}_{t^{\alpha}}=\mathcal{E}_{(\alpha,1)}$ performs well as a knot energy ([10]). In particular, see [2] for (A.3).

Let $\psi(s_{1},s_{2})$ be the angle between \mbox{\boldmath\tau}(s_{1}) and \mbox{\boldmath\tau}(s_{2}). We set

[TABLE]

The following is an extension of the cosine formula for generalized O’Hara energies.

Theorem 2.1

Under (A.1)–(A.4), a generalized cosine formula

[TABLE]

holds. Here $\mathrm{p.v.}\iint=\lim_{\varepsilon\to+0}\iint_{|s_{1}-s_{2}|\geqq\varepsilon}$ is the integration in the principal value sense. 2. 2.

In addition to (A.1)–(A.4), if we assume (A.5) (b), then

[TABLE]

can be defined, and it holds that

[TABLE]

We do not need (A.5) (a). Proof of this will be given in the next section.

Abrams-Cantarella-Fu-Ghomi-Howard [1] established a condition under which the right circle minimizes the knot energies. Our theorem establishes another type of condition. Assume (A.1)–(A.4) and (A.5) (b). Set

[TABLE]

Let $C$ be a circle with a circumstance that is the same as the total length $\mathcal{L}$ of \mathrm{Im}\mbox{\boldmathf}. As the conformal angle for $C$ vanishes identically, we have $\mathcal{E}_{\Phi,3}(C)=0$. Since the energy density of $\mathcal{E}_{\Phi,3}$ is non-negative, we have

[TABLE]

Consequently we obtain

Corollary 2.1

Assume (A.1)–(A.4), and (A.5) (b), If $C$ is a minimizer of $\mathcal{E}_{\Phi,4}$ under the length-constraint, then it minimizes $\mathcal{E}_{\Phi}$ under the constraint.

In the case of O’Hara $\Phi(t)=t^{\alpha}$, the function $\Theta_{\Phi}$ is a constant, and the assertion of Theorem 2.1 becomes

[TABLE]

This coincides with (1) when $\alpha=2$. Now consider the normalized O’Hara energy:

[TABLE]

It is scale-invariant. The quantities

[TABLE]

are Möbius invariant (for the last one, see [4, 6]), but not

[TABLE]

Consequently, we can see from our formula how far the energy is from the Möbius invariant property.

3 Proof

Employing the argument in [7], we can prove Theorem 2.1 very simply. In [7], we proved

[TABLE]

under (A.1)–(A.4). We can see that the conformal angle is given as

[TABLE]

Hence we have

[TABLE]

Using

[TABLE]

we obtain the first assertion of Theorem 2.1. Indeed, the integrand of (4) is

[TABLE]

If we also assume (A.5) (b), then

[TABLE]

Therefore, the integrand

[TABLE]

is non-negative, and

[TABLE]

Hence, the integration in the principal value sense becomes one in the usual $L^{1}$ sense. Furthermore, the angle $\varphi_{{}_{\Phi}}$ is defined. This shows the second assertion of Theorem 2.1.