In this paper, we study the evolving behaviors of the first eigenvalue of the Laplace- Beltrami operator under the normalized backward Ricci flow, construct various quantities which are monotonic under the backward Ri...In this paper, we study the evolving behaviors of the first eigenvalue of the Laplace- Beltrami operator under the normalized backward Ricci flow, construct various quantities which are monotonic under the backward Ricci flow and get upper and lower bounds. We prove that in cases where the backward Ricci flow converges to a sub-Riemannian geometry after a proper rescaling, the eigenvalue evolves toward zero.展开更多
基金Supported by NSFC(Grant No.11001268)Chinese Universities Scientific Fund(Grant No.2014QJ002)
文摘In this paper, we study the evolving behaviors of the first eigenvalue of the Laplace- Beltrami operator under the normalized backward Ricci flow, construct various quantities which are monotonic under the backward Ricci flow and get upper and lower bounds. We prove that in cases where the backward Ricci flow converges to a sub-Riemannian geometry after a proper rescaling, the eigenvalue evolves toward zero.
