2024/03/06 by Pak-Yeung Chan, Chan, Pak-Yeung, Ronan J. Conlon +3 · 3 citations
Biochemistry, Genetics and Molecular Biology · Physics and Astronomy · #53E20 #53E30 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #Endoplasmic Reticulum Stress and Disease #FOS: Mathematics #Lipid metabolism and biosynthesis
paper · pdf · doi:10.48550/arxiv.2403.04089
openalex publication_date 2024/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In 1996, H.-D. Cao constructed a U(n)-invariant steady gradient Kähler-Ricci soliton on ℂn and asked whether every steady gradient Kähler-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 Kähler-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 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 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.