On Otsuki tori and their Willmore energy (Q448235)

An Otsuki torus [\textit{T. Otsuki}, Am. J. Math. 92, 145--173 (1970; Zbl 0196.25102)] is a compact, minimal and embedded hypersurface in \(\mathbb{S}^3 \subset \mathbb{R}^4\) that can be parametrized by NEWLINE\[NEWLINE \begin{aligned} x &= \sqrt{1-h^2-(h')^2} \cos\alpha, \\ y &= \sqrt{1-h^2-(h')^2} \sin\alpha, \\ u &= h(\theta)\sin\theta + h'(\theta)\cos\theta, \\ v &= h'(\theta)\sin\theta - h(\theta)\cos\theta \end{aligned} NEWLINE\]NEWLINE where \(h(\theta)\) is a (necessarily periodic) solution to a certain differential equation with a rationality constraint on its minimal period. Alternative definitions are conceivable as well. Otsuki tori (with the single exception of the Clifford torus) are in a natural one-to-one correspondence with rational numbers in the interval \((1/2, \sqrt{2}/2)\). If \(p/q\) is a reduced rational number in this interval, the corresponding Otsuki torus is denoted by \(O_{p/q}\).NEWLINENEWLINEThe authors' main result is the estimate \(4 \pi q < W(O_{p/q}) < \sqrt{2} \pi^2 q\) for the Willmore energy \(W(O_{p/q})\) of Otsuki tori. If the torus is invariant under the antipodal map, the lower bound can be improved to \(W(O_{p/q}) > 32\pi\). These results show that the Willmore conjecture holds for Otsuki tori.