We provide explicit formulae for the first eigenvalue of Laplace-Beltrami operator on a compact rank one symmetric space (CROSS) endowed with any homogeneous metric. As consequences, we prove that metrics CROSSes are isospectral if and only they isometric, also discuss their stability (or lack thereof) as solutions to Yamabe problem.

We study first nonzero eigenvalues for the p p -Laplacian on Kähler manifolds. Our result is a lower bound closed (Neumann) eigenvalue of comp...