具有平行第二基本形式的水平超曲面的Ricci孤立子
On Ricci solitons whose level hypersurfaces have parallel second fundamental form
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中文总结 AI 辅助
该研究刻画了水平超曲面具平行第二基本形式的梯度Ricci孤立子的局部结构，证明相关刚性结论并验证Cao猜想在特定情形下成立。
中文摘要 AI 辅助
我们研究势函数的水平超曲面具有平行第二基本形式的梯度Ricci孤立子。我们证明这类孤立子局部上是一维基与爱因斯坦纤维的多重扭曲积，且势函数仅依赖于基。因此，局部而言，该条件刻画了Ricci孤立子诸多经典构造中出现的多重扭曲积结构。我们进一步证明，若任意梯度Ricci孤立子的一个正则水平超曲面具有平行第二基本形式和平行内在Ricci张量，则其具有相同的局部结构。作为应用，我们证明，只要一个正则水平超曲面具有该性质，标量曲率恒定的完备非稳态梯度Ricci孤立子就是刚性的；特别地，对于具有该性质的梯度收缩Ricci孤立子，Cao猜想成立。我们还证明，正则水平超曲面具有平行第二基本形式且至多一个非零主曲率的完备梯度收缩Ricci孤立子是刚性的，无需对标量曲率作任何假设。
英文摘要
We study gradient Ricci solitons whose potential functions have level hypersurfaces with parallel second fundamental form. We show that such a soliton is locally a multiply warped product of a one-dimensional base and Einstein fibers, with potential function depending only on the base. Thus, locally, this condition characterizes the multiply warped structure that occurs in many classical constructions of Ricci solitons. We further prove that the same local structure follows for any gradient Ricci soliton if just one regular level hypersurface has parallel second fundamental form and parallel intrinsic Ricci tensor. As an application, we prove that a complete nonsteady gradient Ricci soliton with constant scalar curvature is rigid as soon as one regular level hypersurface has this property. In particular, the conjecture of Cao holds for gradient shrinking Ricci solitons with this property. We also prove that a complete gradient shrinking Ricci soliton whose regular level hypersurfaces have parallel second fundamental form and at most one nonzero principal curvature is rigid, without any assumption on the scalar curvature.
发表机构
- Universidade de Brasília（巴西利亚大学）
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