Appendix A On extending the conjecture of Benguria and Loss

Proof. Consider \(\sigma_\tau\) to be the curve in \(\mathcal{D}_+^{2,1}\) which has \(\sigma_\tau(\mathbf{e}_1)=\mathbf{e}_1\), \(L(\sigma_\tau)=2\pi\) and induced diffeomorphism \(\phi_{\sigma_\tau}=\psi_\tau\) where \(\psi_\tau\) is given by (4.2). Note, that for \(\tau\neq 0\), \(\sigma_\tau\) is not closed. One computes that for \(f_\tau=\kappa_{\sigma_\tau}^{-1/2}\in C^{0,1}(\mathbb{S}^1)\subset H^1(\mathbb{S}^1)\), that \(\mathcal{E}_S[\sigma_\tau,f_\tau]=\mathcal{E}_G[\sigma_\tau]=0\). However, \[L_{\sigma_\tau}f_\tau=-f_\tau'' +\kappa_{\sigma_\tau}^2 f_\tau = f_\tau +C(\tau)\delta_{\mathbf{e}_2}-C(\tau)\delta_{-\mathbf{e}_2},\] distributionally and the constant \(C(\tau)\neq 0\) if and only if \(\tau\neq 0\). Hence, for \(\tau\neq 0\), \(f_\tau\) is not an eigenfunction and so there must be a \(\hat{f}_\tau\in C^2(\mathbb{S}^1)\) with \(\mathcal{E}_S[\sigma_\tau,\hat{f}_\tau]<0\). Consider the elements \(\psi_{\tau}^\lambda\in \mathrm{ Diff}_+^{1,1}(\mathbb{S}^1)\) given by (4.3) and pick \(\sigma_{\tau}^\lambda\in \mathcal{D}_+^{2,1}\) so that \(\sigma_\tau^\lambda(\mathbf{e}_1)=\mathbf{e}_1\), \(L(\sigma_\tau^\lambda)=2\pi\) and the induced diffeomorphism is \(\psi_{\tau}^\lambda\). Clearly, \(\sigma_{\tau}^\lambda\to \sigma_\tau\) as \(\lambda\to 1\) in the \(C^2\) topology. Hence, \(\mathcal{E}_S[\sigma_{\tau}^{\lambda},\hat{f}_\tau]\to \mathcal{E}_S[\sigma_{\tau},\hat{f}_\tau]\) as \(\lambda\to 1\). Hence, for \(\tau\neq 0\) and \(\lambda>1\) sufficiently close to \(1\), we obtain a \(\sigma\in \mathcal{D}_+^\infty\) with \(n_B(\sigma)=\infty\) and \(\mathcal{E}_S[\sigma, \hat{f}_\tau]<0\) by smoothing out \(\sigma_{\tau}^\lambda\) as in Lemma 4.7. Smoothing out \(\hat{f}_\tau\) gives \(f\) so that \(\mathcal{E}_S[\sigma,f]<0\).