Eigenvalue monotonicity for the Ricci-Hamilton flow, revised version
  This is a revised version of our short note [arxiv.math.DG/0403065] where we discuss the monotonicity of the eigen-values of the Laplacian operator to the Ricci-Hamilton flow on a compact or a complete non-compact Riemannian manifold. We show that the eigenvalue of the Lapacian operator on a compact domain associated with the evolving Ricci flow is non-decreasing provided the scalar curvature having a non-negative lower bound and Einstein tensor being not too negative. This result will be useful in the study of blow-up models of the Ricci-Hamilton flow. We remark that using our idea, D.Kokotov and D.Korotkin [Normalized Ricci Flow on Riemann Surfaces and Determinant of Laplacian, Letters in Mathematical Physics, 71(3)(2005)241-242] can give a simple proof of the fact that the determinant of Laplace operator in a smooth metric over compact Riemann surfaces of an arbitrary genus g monotonously grows under the normalized Ricci flow. Together with results of Hamilton and B.Chow that under the action of the normalized Ricci flow a smooth metric tends asymptotically to the metric of constant curvature, this leads to a simple proof of the Osgood-Phillips-Sarnak theorem stating that within the class of smooth metrics with fixed conformal class and fixed volume the determinant of the Laplace operator is maximal on the metric of constant curvature. One may consider similar problems for other geometric flow and other elliptic operators, such as Yamabe flow, or Kahler-Ricci flow. 
Li Ma