Evolution of laplacian spectrum under Hamilton's Ricci flow

Abstract

Spectrum of the Laplacian-Beltrami operator on closed Riemannian manifolds were extensively studied by geometric analysts since 1970s. The results obtained by S.Y. Cheng, Peter Li and S.T. Yau revealed many beautiful relations between the Laplacian spectrum and the geometric and topological structures of manifolds. It is an interesting idea to trace out how the Laplacian eigenvalues evolve under the Ricci flow, a geometric evolution which serves primarily in understanding the topology of manifolds. We were able to prove that under the Ricci flow equation ∂/∂t*gij = −2Rij on 3-manifolds with positive Ricci curvature, and also n-manifolds (n ≥ 4) with positive curvature operator, all the Laplacian eigenvalues λα(g(t)), α = 0, 1, 2, 3, · · · are monotonically increasing after some finite time. Furthermore, we are able to derive a first derivative estimate for Laplacian eigenvalues under the normalized Ricci flow equation ∂/∂t*gij = −2Rij + 2r/n*gij. We have a high expectation that these results will give rise to a better understanding of the asymptotic behavior of the Ricci flow.

Date of Award	2007
Original language	English
Awarding Institution	The Hong Kong University of Science and Technology