一类特殊的拟几乎Einstein度量直径的下界估计(英文)

Lower diameter estimate for a special quasi-almost-Einstein metric

加权Myer型定理给出了具有带正下界的τ-Bakry-Emery曲率的完备黎曼流形直径的上界估计,紧致流形直径的下界估计也是有趣的问题.本文首先运用Hopf极大值原理证明了一类特殊的τ-拟几乎Einstein度量势函数的梯度估计.运用该梯度估计得到了该度量直径的下界估计.该结果推广了王林峰的关于紧致下-拟Einstein度量直径下界估计的结果.

The weighted Myers＇ theorem gives an upper bound estimate for the diameter of a complete Riemannian manifold with the τ-Bakry-Emery curvature bounded from below by a positive number. The lower bound estimate for the diameter of a compact manifold is also an interesting question. In this paper, a gradient estimate for the potential function of a special τ-quasi-almost-Einstein metric was established by using the Hopf＇s maximum principle. Based on it, a lower bound estimate for the diameter of this metric was derived. The result generalizes Wang＇s lower diameter estimate for compact r-quasi-Einstein metrics.