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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. 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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3. Steady gradient Kähler-Ricci solitons on crepant resolutions of Calabi-Yau cones

   Conlon, Ronan J.; Deruelle, Alix (June 2020, ArXivorg)

   We show that, up to the flow of the soliton vector field, there exists a unique complete steady gradient Kähler-Ricci soliton in every Kähler class of an equivariant crepant resolution of a Calabi-Yau cone converging at a polynomial rate to Cao's steady gradient Kähler-Ricci soliton on the cone. 

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4. 5-dimensional space-periodic solutions of the static vacuum Einstein equations

   https://doi.org/10.1007/JHEP12(2020)002  

   Khuri, Marcus; Weinstein, Gilbert; Yamada, Sumio (December 2020, Journal of High Energy Physics)

   null (Ed.)

   A bstract An affirmative answer is given to a conjecture of Myers concerning the existence of 5-dimensional regular static vacuum solutions that balance an infinite number of black holes, which have Kasner asymptotics. A variety of examples are constructed, having different combinations of ring S 1 × S 2 and sphere S 3 cross-sectional horizon topologies. Furthermore, we show the existence of 5-dimensional vacuum solitons with Kasner asymptotics. These are regular static space-periodic vacuum spacetimes devoid of black holes. Consequently, we also obtain new examples of complete Riemannian manifolds of nonnegative Ricci curvature in dimension 4, and zero Ricci curvature in dimension 5, having arbitrarily large as well as infinite second Betti number. 

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5. Propagation of symmetries for Ricci shrinkers

   https://doi.org/10.1515/ans-2022-0071  

   Colding, Tobias Holck; Minicozzi II, William P. (January 2023, Advanced Nonlinear Studies)

   Abstract We will show that if a gradient shrinking Ricci soliton has an approximate symmetry on one scale, this symmetry propagates to larger scales. This is an example of the shrinker principle which roughly states that information radiates outwards for shrinking solitons. 

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