Conjugate points along spherical harmonics (Q6638283)

The paper is devoted to studying the existence of conjugate points along solutions of the quasi-geostrophic equations on the \(2\)-dimensional sphere \(\mathbb S^2\).\N\NConsider the classical spherical harmonics \(Y_{l_1m_1}= C_l^mP_l^{|m|}(\mu) e^{im\lambda}\) those are the eigenfunctions of the Laplacian, \[\Delta Y_{lm}= -l(l+1)Y_{lm}, 1< m_1 \leq l_1\] where the coefficient \[C_l^m = (-1)^m\sqrt{\frac{2l+1}{4\pi}\frac{(l-|m|)!}{(l+|m|)!}}\] and the associated Legendre polynomial is \[P_l^{|m|}(\mu) = \frac{(1-\mu^2)^{m/2}}{2^ll!}\frac{d^{l+|m|}}{d\mu^{l+|m|}}(\mu^2-1)^l.\] The main result of the paper is the following:\N\N\textit{Theorem 3.4.} \begin{enumerate}\N\item The Misiolek criterion \(MC(e_{l_1m_1}, e_{m-m})\) \(= \langle \Delta\{e_{l_1m_1}, e_{m-m}\}, \{e_{l_1m_1}, e_{m-m}\}\rangle - \langle \{\Delta e_{l_1m_1}, e_{m-m}\}, \{ e_{l_1m_1}, e_{m-m}\} \rangle >0\) for all \(2\leq m \leq m_1\) and the symplectic dual \(e_{l_1m_1} =\nabla^\perp Y_{l_1m_1}\) with respect to the symplectic gradient \(\perp=\mathrm{sgrad}\) on the divergence-free vector fields Lie algebra \(\mathfrak g = T_e\mathcal D^s_{\mathrm{vol}}(\mathbb S^2)\) of the diffeomorphisms conserving the volume;\N\N\item \(MC(e_{l_11}, e_{l_21}) >0\) for any \(3\leq l_1, 2\leq l_2 <l_1\). The impact of the Coriolis force on the occurrence of conjugate points is also investigated: the Coriolis effect stabilizes the system in generating conjugate points (Proposition 4.3, Corollary 4.4). \end{enumerate}