On higher order eigenvalues of the spherical Laplacian operator (Q812882)

The author studies the eigenvalues of the Laplace operator on spherical bands. Let \(S^{2}\) be the unit sphere in \(\mathbb R^{3}\) and let \(\Delta_{S^{2}}\) stand for the spherical Laplacian on \(S^{2}\). We denote the Euler coordinates on \(S^{2}\) by \(X(\theta,\phi)=(\cos\theta\sin\phi,\sin\theta\sin\phi,\cos\phi)\), where \(0\leq\phi\leq\pi\), \(0\leq\theta<2\pi\). We fix \(0<\phi_{0}<\phi_{1}<\pi\). Put \[ B(s)=\{X(\theta,\phi)|\qquad \phi_{0}+s<\phi<\phi_{1}+s,\quad 0\leq\theta<2\pi\} \] for \(0\leq s\leq\pi-\phi_{1}\). By \(\nu_{2}(s)\) we denote the second eigenvalue of the following problem. \[ \Delta_{S^{2}}u+\lambda u=0\quad\text{in}\quad B(s),\qquad u=0\quad\text{on}\quad\partial B(s). \] The main result of this paper is as follows. The function \(\nu_{2}(s)\) is decreasing in \((0, (\phi_{1}-\phi_{0})/2)\) and attains its minimum when the band \(B(s)\) is symmetric with respect to the equator. This result extends the previous result of the author, Theorem 3.1 in [Proc. Am. Math. Soc. 132, No. 1, 157--164 (2004; Zbl 1037.35049)].