The Sasaki–Ricci Flow and Compact Sasaki Manifolds of Positive Transverse Holomorphic Bisectional Curvature
The Sasaki–Ricci Flow and Compact Sasaki Manifolds of Positive Transverse Holomorphic Bisectional Curvature
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DOI：
10.1007/s12220-012-9311-7
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发表时间：
2011-03
影响因子：
1.1
通讯作者：
Weiyong He
中科院分区：
文献类型：
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作者：
Weiyong He
我们证明了Kähler流形上的Perelman泛函在Sasaki流形上有一个自然的对应。利用这个泛函，我们证明了Perelman关于Kähler-Ricci流（第一类为正）的结果可以推广到Sasaki-Ricci流，包括流的直径和沿沿着方向的数量曲率的一致界.利用Bando和Mok的方法和Kähler-Ricci流的结果，我们还证明了横对分曲率的正性沿沿着Sasaki-Ricci流是保持的.特别地，我们证明了Sasaki-Ricci流收敛到Sasaki-Ricci孤子时的初始度量具有非负的横截二分曲率。
We show that Perelman’sfunctional on Kähler manifolds has a natural counterpart on Sasaki manifolds. We prove, using this functional, that Perelman’s results on Kähler–Ricci flow (the first Chern class is positive) can be generalized to Sasaki–Ricci flow, including the uniform bound on the diameter and the scalar curvature along the flow. We also show that positivity of transverse bisectional curvature is preserved along Sasaki–Ricci flow, using Bando and Mok’s methods and results in Kähler–Ricci flow. In particular, we show that the Sasaki–Ricci flow converges to a Sasaki–Ricci soliton when the initial metric has nonnegative transverse bisectional curvature.