一族Kähler飞翼稳态Ricci孤立子
A family of Kähler flying wing steady Ricci solitons
AI总结：
本文构造了$\mathbb{C}^{n}$（$n\geq3$）上一族非$U(n)$不变的$U(1)\times U(n-1)$不变稳态梯度Kähler-Ricci孤立子，具有正曲率算子，否定回答了Cao的问题，并给出Lai飞翼构造的Kähler类比。
摘要：
1996年，H.-D. Cao在$\mathbb{C}^{n}$上构造了一个$U(n)$不变的稳态梯度Kähler-Ricci孤立子，并问$\mathbb{C}^{n}$上每个具有正曲率的稳态梯度Kähler-Ricci孤立子是否必然$U(n)$不变（从而在缩放意义下唯一）。最近，Apostolov-Cifarelli对$n=2$给出了否定回答。本文中，我们对$n\geq3$，在$\mathbb{C}^{n}$上构造了一族$U(1)\times U(n-1)$不变但非$U(n)$不变的完备稳态梯度Kähler-Ricci孤立子，其实$(1,1)$型上的曲率算子严格正（特别地，具有严格正的截面曲率），从而对$n\geq3$否定了Cao的问题。这族稳态Ricci孤立子在Cao的$U(n)$不变稳态Kähler-Ricci孤立子与雪茄孤立子和Cao的$U(n-1)$不变稳态Kähler-Ricci孤立子的乘积之间插值。这提供了Lai的黎曼飞翼构造的Kähler类比。在证明过程中，我们还证明了赋予Fubini-Study度量的$\mathbb{P}^{n}$的几乎直径刚性即使在实$(1,1)$型上曲率算子以$2$为下界时也不成立。
英文摘要：
In $1996$, H.-D. Cao constructed a $U(n)$-invariant steady gradient Kähler-Ricci soliton on $\mathbb{C}^{n}$ and asked whether every steady gradient Kähler-Ricci soliton of positive curvature on $\mathbb{C}^{n}$ is necessarily $U(n)$-invariant (and hence unique up to scaling). Recently, Apostolov-Cifarelli answered this question in the negative for $n=2$. Here, we construct a family of $U(1)\times U(n-1)$-invariant, but not $U(n)$-invariant, complete steady gradient Kähler-Ricci solitons with strictly positive curvature operator on real $(1,\,1)$-forms (in particular, with strictly positive sectional curvature) on $\mathbb{C}^{n}$ for $n\geq3$, thereby answering Cao's question in the negative for $n\geq3$. This family of steady Ricci solitons interpolates between Cao's $U(n)$-invariant steady Kähler-Ricci soliton and the product of the cigar soliton and Cao's $U(n-1)$-invariant steady Kähler-Ricci soliton. This provides the Kähler analog of the Riemannian flying wings construction of Lai. In the process of the proof, we also demonstrate that the almost diameter rigidity of $\mathbb{P}^{n}$ endowed with the Fubini-Study metric does not hold even if the curvature operator is bounded below by $2$ on real $(1,\,1)$-forms.
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