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1. A family of Kähler flying wing steady Ricci solitons

   Chan, Pak-Yeung; Conlon, Ronan; Lai, Yi (March 2024, arXivorg)

   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. 

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2. On finite time Type I singularities of the Kähler–Ricci flow on compact Kähler surfaces

   https://doi.org/10.4171/JEMS/1485  

   Cifarelli, Charles; Conlon, Ronan J.; Deruelle, Alix (July 2024, Journal of the European Mathematical Society)

   We show that the underlying complex manifold of a complete non-compact two-dimensional shrinking gradient Kähler-Ricci soliton (M,g,X) with soliton metric g with bounded scalar curvature Rg whose soliton vector field X has an integral curve along which Rg↛0 is biholomorphic to either C×P1 or to the blowup of this manifold at one point. Assuming the existence of such a soliton on this latter manifold, we show that it is toric and unique. We also identify the corresponding soliton vector field. Given these possibilities, we then prove a strong form of the Feldman-Ilmanen-Knopf conjecture for finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces, leading to a classification of the bubbles of such singularities in this dimension. 

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3. O(2)-symmetry of 3D steady gradient Ricci solitons

   https://doi.org/10.2140/gt.2025.29.687  

   Lai, Yi (January 2025, Geometry & Topology)

   We prove that all 3D steady gradient Ricci solitons are O(2)-symmetric. The O(2)-symmetry is the most universal symmetry in Ricci flows with any type of symmetries. Our theorem is also the first instance of symmetry theorem for Ricci flows that are not rotationally symmetric. We also show that the Bryant soliton is the unique 3D steady gradient Ricci soliton with positive curvature that is asymptotic to a ray. 

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4. An Aubin continuity path for shrinking gradient Kähler–Ricci solitons

   https://doi.org/10.1515/crelle-2024-0053  

   Cifarelli, Charles; Conlon, Ronan J.; Deruelle, Alix (August 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Let 𝐷 be a toric Kähler–Einstein Fano manifold. We show that any toric shrinking gradient Kähler–Ricci soliton on certain toric blowups of C×D satisfies a complex Monge–Ampère equation. We then set up an Aubin continuity path to solve this equation and show that it has a solution at the initial value of the path parameter. This we do by implementing another continuity method. 

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5. The existence of the Kähler–Ricci soliton degeneration

   https://doi.org/10.1017/fmp.2023.5  

   Blum, Harold; Liu, Yuchen; Xu, Chenyang; Zhuang, Ziquan (January 2023, Forum of Mathematics, Pi)

   Abstract We prove an algebraic version of the Hamilton–Tian conjecture for all log Fano pairs. More precisely, we show that any log Fano pair admits a canonical two-step degeneration to a reduced uniformly Ding stable triple, which admits a Kähler–Ricci soliton when the ground field . 

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