Lower diameter estimate for a special quasi-almost-Einstein metric
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摘要: 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.
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关键词:
- 拟几乎~Einstein~度量 /
- 梯度估计 /
- 直径估计
Abstract: 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.-
Key words:
- quasi-almost-Einstein metric /
- gradient estimate /
- diameter estimate
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