On the eigenvalues of the Laplacian for certain perturbations of the standard Euclidean metric on \(S^{2}\) (Q1049675)

Let \(\lambda_i(g)\) be the \(i\)th eigenvalue of the scalar Laplacian \(\Delta_g\) on a compact Riemannian manifold \((M,g)\) of dimension \(n\). Let \(g_0\) be the standard round metric on the unit sphere. Lichnerowicz showed that if \(\text{Ric}_g\geq(n-1)g\), then \(\lambda_1(g)\geq \lambda_1(g_0)\). In general, it is not true that \(\lambda_i(g)\geq\lambda_i(g_0)\) for \(i>1\). However, the author shows that if \(M=S^2\) and if \(j\) is fixed, then \(\lambda_i(g)\geq\lambda_i(g_0)\) for all \(i\leq j\) for metrics sufficiently close to \(g_0\) that are obtained by analytically perturbing \(g_0\) through rotationally symmetric conformal metrics.