Eigenvalues under the backward Ricci flow on locally homogeneous closed 3-manifolds

摘要

In this paper, we study the evolving behaviors of the first eigenvalue ofLaplace-Beltrami operator under the normalized backward Ricci flow, constructvarious quantities which are monotonic under the backward Ricci flow and getupper and lower bounds. We prove that in cases where the backward Ricci flowconverges to a sub-Riemannian geometry after a proper rescaling, the eigenvalueevolves toward zero.