VARIATION OF THE FIRST EIGENVALUE OF p-LAPLACIAN ON EVOLVING GEOMETRY AND APPLICATIONS

Let (M, g) be an n-dimensional compact Riemannian manifold whose metric g(t) evolves by the generalised abstract geometric flow. This paper discusses the variation formulas, monotonicity and differentiability for the first eigenvalue of the p-Laplacian on (M,g(t)) . It is shown that the first nonzero eigenvalue is monotonically nondecreasing along the flow under certain geometric conditions and that it is differentiable almost everywhere. These results provide a unified approach to the study of eigenvalue variations and applications under many geometric flows.