—类特殊的拟几乎Einstein度量直径的下界估计(英)简

加权~Myer~型定理给出了具有带正下界的~$\tau$-Bakry-\'{E}mery~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~$\tau$-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~$\tau$-拟~Einstein~度量直径下界估计的结果.

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

The weighted Myers' theorem gives an upper bound estimate
for the diameter of a complete Riemannian manifold with the
$\tau$-Bakry-\'{E}mery 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
$\tau$-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 $\tau$-quasi-Einstein
metrics.