Eigenvalues on spheres

For every smooth Riemannian metric on the two sphere whose Gaussian curvature is bounded below by one, we prove that each positive Laplace eigenvalue, counted with multiplicity, is no smaller than the corresponding eigenvalue of the unit round sphere. Equality at any positive position in the ordered spectrum forces the metric to be isometric to the unit round metric. We further establish a sharp finite spectral counting comparison for Alexandrov two spheres with curvature bounded below by one. At every positive spectral threshold of the unit round sphere, the number of Laplace eigenvalues below or at that threshold, counted with multiplicity, does not exceed the corresponding number for the round sphere. Equality at any such threshold forces the Alexandrov sphere to be isometric to the unit round sphere. As an application, we obtain the sharp Euclidean dimension bound for spaces of polynomial growth harmonic functions on complete three dimensional manifolds with nonnegative sectional curvature and positive asymptotic volume ratio, together with rigidity in the equality case.