A family of K\"ahler flying wing steady Ricci solitons

In $1996$, H.-D. Cao constructed a $U(n)$-invariant steady gradient K\"ahler-Ricci soliton on Cⁿ and asked whether every steady gradient K\"ahler-Ricci soliton of positive curvature on Cⁿ 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)× U(n-1)-invariant, but not $U(n)$-invariant, complete steady gradient K\"ahler-Ricci solitons with strictly positive curvature operator on real $(1,\,1)$-forms (in particular, with strictly positive sectional curvature) on Cⁿ for n≥3, thereby answering Cao's question in the negative for n≥3. This family of steady Ricci solitons interpolates between Cao's $U(n)$-invariant steady K\"ahler-Ricci soliton and the product of the cigar soliton and Cao's $U(n-1)$-invariant steady K\"ahler-Ricci soliton. This provides the K\"ahler analog of the Riemannian flying wings construction of Lai. In the process of the proof, we also demonstrate that the almost diameter rigidity of Pⁿ endowed with the Fubini-Study metric does not hold even if the curvature operator is bounded below by $2$ on real $(1,\,1)$-forms.