3 Symmetries of the functionals

Consider the group homomorphism \(\Gamma: \mathrm{SL}(2,\mathbb R)\to \mathrm{ Diff}^\infty_+(\mathbb{S}^1)\) given by \[\Gamma(L) = \mathbf{x}\mapsto \frac{L\cdot \mathbf{x}}{|L\cdot \mathbf{x}|}\] where \(\mathbf{x}\in \mathbb{S}^1\) and \(L\in \mathrm{SL}(2,\mathbb R)\). Denote the image of \(\Gamma\) by \(\mathrm{ M\ddot{o}b}(\mathbb{S}^1)\) which we refer to as the Möbius group of \(\mathbb{S}^1\). One computes that \[\mathbf{T}_0\circ \Gamma(L)=\frac{ L \cdot \mathbf{T}_0}{|L \cdot \mathbf{T}_0|}.\] These are precisely the unit tangent maps of the ovals of [1]. That is, \[\mathcal{O}=\left\{\sigma\in \mathcal{D}_+^\infty: \phi_\sigma\in \mathrm{ M\ddot{o}b}(\mathbb{S}^1)\right\}.\]

3.1 The Schwarzian derivative

For \(\phi\in \mathrm{ Diff}_+^3(\mathbb{S}^1)\) the Schwarzian derivative of \(\phi\) is defined as \[S_{\theta}(\phi)=\frac{\phi^{\prime\prime\prime}}{\phi^{\prime}}-\frac{3}{2}\left(\frac{\phi^{\prime\prime}}{\phi^{\prime}}\right)^2,\] where primes denote derivatives with respect to \(\theta\). A fundamental feature of the Schwarzian derivative is that it satisfies the following co-cycle property \[\tag{3.1} S_\theta(\phi\circ \psi)=\left(S_\theta(\phi)\circ \psi\right)\cdot (\psi^{\prime})^2+S_{\theta}(\psi),\] where \(\phi,\psi\in \mathrm{ Diff}_+^3(\mathbb{S}^1)\). After some computation, one verifies that the Schwarzian derivative gives the following intrinsic characterization of \(\mathrm{ M\ddot{o}b}(\mathbb{S}^1)\) \[\tag{3.2} \phi \in \mathrm{ M\ddot{o}b}(\mathbb{S}^1) \iff S_\theta(\phi)+2(\phi')^2-2 =0.\]

The Schwarzian derivative arises most naturally in the context of projective differential geometry. This perspective also gives a conceptual proof of (3.2). For this proof as well as the necessary background the reader may consult the Appendix B as well as the references cited there.

3.2 Projective Symmetries

We now describe the natural symmetries of (1.2), (1.3) and (1.4). We also introduce a notion of duality for strictly convex degree-one curves – this duality will streamline some of the arguments. For \(\sigma\in \mathcal{D}^{k+1,\alpha}\), define the dual curve, \(\sigma^*\in \mathcal{D}^{k+1,\alpha}\) to be the unique curve with \[\begin{array}{cccc} \phi_{\sigma^*}=\phi_{\sigma}^{-1}, & \mathbf{x}_{\sigma^*}= \mathbf{x}_{\sigma} & \mbox{and} & L(\sigma^*)={L(\sigma)}.\end{array}\] That is, \(\sigma^*\) is the curve whose induced diffeomorphism is the inverse to the induced diffeomorphism of \(\sigma\). Clearly, \((\sigma^*)^*=\sigma\). To proceed further, we note that the functionals (1.3) and (1.4) can, by integrating by parts, be made to naturally involve the Schwarzian derivative. To see this fix \(\sigma \in \mathcal{D}_+^{4}\) with \(L(\sigma)=2\pi\). As \(\kappa_\sigma=\phi_\sigma'\), \[\tag{3.3} \begin{aligned} \mathcal{E}_G[\sigma] &=\int_{\mathbb{S}^1} \frac{(\phi_\sigma'')^2}{4(\phi_\sigma')^3} -\frac{1}{\phi_\sigma'}+\phi_\sigma' \; \mathrm{d}\theta \\ &=\int_{\mathbb{S}^1} \left( \frac{\phi_\sigma''}{2(\phi_\sigma')^3}\right)' \phi_\sigma'+\frac{3(\phi_\sigma'')^2}{4(\phi_\sigma')^3} -\frac{1}{\phi_\sigma'}+\phi_\sigma' \; \mathrm{d}\theta \\ &=\frac{1}{2} \int_{\mathbb{S}^1} \frac{S_{\theta}(\phi_\sigma) +2(\phi_\sigma')^2-2}{\phi_\sigma'} \; \mathrm{d}\theta, \end{aligned}\] where the second equality follows by integrating by parts. Likewise, \[\tag{3.4} \mathcal{E}_G^*[\sigma] =-\frac{1}{2} \int_{\mathbb{S}^1} S_{\theta}(\phi_\sigma) +2(\phi_\sigma')^2-2 \; \mathrm{d}\theta.\] An immediate consequence of this is the following useful fact,

We now may define the desired actions. Consider first the right action of \(\mathrm{ M\ddot{o}b}(\mathbb{S}^1)\) on \(\mathcal{D}^{k+1,\alpha}\), \(\sigma\cdot \varphi = \sigma'\) where \(\sigma'\in \mathcal{D}^{k+1,\alpha}\) is the unique element with \[\begin{array}{cccc} \mathbf{T}_{\sigma'}=\mathbf{T}_{\sigma}\circ \varphi, & \mathbf{x}_{\sigma'}= \mathbf{x}_{\sigma} & \mbox{and} & L(\sigma')=L(\sigma).\end{array}\] Notice, that if \(\sigma\in \mathcal{D}^{k+1,\alpha}_+\) is strictly convex, then so is \(\sigma'\) and in this case we have that \(\phi_{\sigma'}=\phi_{\sigma}\circ \varphi\). With this in mind, we also consider a left action of \(\mathrm{ M\ddot{o}b}(\mathbb{S}^1)\) on \(\mathcal{D}^{k+1,\alpha}_+\), \(\varphi\cdot \sigma =\sigma'\), where \(\sigma'\in \mathcal{D}^{k+1,\alpha}\) is the unique element with \[\begin{array}{cccc} \phi_{\sigma'}=\varphi \circ \phi_{\sigma}, & \mathbf{x}_{\sigma'}= \mathbf{x}_{\sigma} & \mbox{and} & L(\sigma')=L(\sigma).\end{array}\] We observe that for \(\sigma \in \mathcal{D}^{k+1}_+\) and \(\varphi\in \mathrm{ M\ddot{o}b}(\mathbb{S}^1)\), \[\varphi \cdot \sigma^*= (\sigma \cdot \varphi^{-1})^*.\] Finally, we define a right action of \(\mathrm{ M\ddot{o}b}(\mathbb{S}^1)\) on \(C^{k,\alpha}(\mathbb{S}^1)\) by \[f\cdot \varphi=\left(\varphi'\right)^{-1/2} f\circ\varphi .\] If we use \(\mathrm{d}\theta\) to identify \(C^\infty(\mathbb{S}^1)\) with \(\Omega^{-1/2}(\mathbb{S}^1)\), then this is the natural pull-back action on \(\Omega^{-1/2}(\mathbb{S}^1)\) – the space of weight \(-1/2\) densities (see Appendix B).

Theorem 1.2 is an immediate consequence of Proposition 3.1 and Proposition 3.2 and the fact that \(\mathrm{ M\ddot{o}b}(\mathbb{S}^1)\) is isomorphic to \(\mathrm{SL}(2,\mathbb R)\).

As a final remark, we observe that we may extend the duality operator to \(\mathcal{D}_+^{\infty} \times C^\infty(\mathbb{S}^1)\) and define a natural dual functional to \(\mathcal{E}_S\). Namely, set \[\begin{array}{ccc}(\sigma, f)^*=(\sigma^*, f\circ \phi_\sigma^{-1}) & \mbox{and} & \mathcal{E}_S^*[\sigma, f]=\int_{\sigma} \frac{|\nabla_\sigma f|^2}{\kappa_\sigma} -\kappa_\sigma f^2 +\frac{(2\pi)^2}{L(\sigma)^2} \frac{f^2}{ \kappa_\sigma} \; \mathrm{d}s. \end{array}\] We then have,