Covariant Extremisation of Flavour-Symmetric Jarlskog Invariants and the Neutrino Mixing Matrix

We examine the possibility that the form of the lepton mixing matrix can be determined by extremising the Jarlskog flavour invariants associated, eg. with the commutator ($C$) of the lepton mass matrices. Introducing a strictly covariant approach, keeping masses fixed and extremising the determinant (Tr $C^3/3$) leads to maximal CP violation, while extremising the sum of the $2 \times 2$ principal minors ($-{\rm Tr} C^2/2$), leads to a non-trivial mixing with zero CP violation. Extremising, by way of example, a general linear combination of two CP-symmetric invariants together, we show that our procedures can lead to acceptable mixings and to non-trivial predictions, eg.\ $|U_{e3}| \simeq \sqrt{2}/3 \sqrt{\Delta m_{12}^2/\Delta m_{23}^2} (1-m_{\mu}/m_{\tau})^2 \simeq 0.07$.