Yamabe流下Laplace算子的第一特征值(英文)

The First Eigenvalue of the Laplace Operator under Yamabe Flow

本文得到Yamabe流下拉普拉斯算子的第一特征值的发展方程.我们证明出,在光滑的齐性流形(Mn,g)上,若λ(t)表示拉普拉斯算子的特征值,那么沿着规范化后的Yamabe流,λ(t) = d,而且沿着非规范化的Yambe流,λ(t) = dect,这里d是一个常数,c表示齐性流形的数量曲率.而且作为发展方程的应用,我们得到了一些这个流下单调的几何量.

In this paper,we derive the evolution equation for the first eigenvalue of Laplace operator along Yamahe flow. We prove that on the smooth homogeneous manifold （Mn ,g0）, if λ（t） denotes the first eigenvalue of Laplace operator, then under normalized Yamahe flow, we have λ= d, and we can also get under the unnormalized Yamabe flow,λ（t） == dew , where c is the scalar curvature of smooth homogeneous manifold and d is a constant. Moreover, as applications, we construct some monotonic quantities under this flow.