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1. Ricci flow and diffeomorphism groups of 3-manifolds

   https://doi.org/10.1090/jams/1003  

   Bamler, Richard; Kleiner, Bruce (August 2022, Journal of the American Mathematical Society)

   We complete the proof of the Generalized Smale Conjecture, apart from the case of R P 3 RP^3 , and give a new proof of Gabai’s theorem for hyperbolic 3 3 -manifolds. We use an approach based on Ricci flow through singularities, which applies uniformly to spherical space forms, except S 3 S^3 and R P 3 RP^3 , as well as hyperbolic manifolds, to prove that the space of metrics of constant sectional curvature is contractible. As a corollary, for such a 3 3 -manifold X X , the inclusion Isom ⁡ ( X , g ) → Diff ⁡ ( X ) \operatorname {Isom}(X,g)\rightarrow \operatorname {Diff}(X) is a homotopy equivalence for any Riemannian metric g g of constant sectional curvature. 

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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. Metric flows with neural networks

   https://doi.org/10.1088/2632-2153/ad8533  

   Halverson, James; Ruehle, Fabian (October 2024, Machine Learning: Science and Technology)

   Abstract We develop a general theory of flows in the space of Riemannian metrics induced by neural network (NN) gradient descent. This is motivated in part by recent advances in approximating Calabi–Yau metrics with NNs and is enabled by recent advances in understanding flows in the space of NNs. We derive the corresponding metric flow equations, which are governed by a metric neural tangent kernel (NTK), a complicated, non-local object that evolves in time. However, many architectures admit an infinite-width limit in which the kernel becomes fixed and the dynamics simplify. Additional assumptions can induce locality in the flow, which allows for the realization of Perelman’s formulation of Ricci flow that was used to resolve the 3d Poincaré conjecture. We demonstrate that such fixed kernel regimes lead to poor learning of numerical Calabi–Yau metrics, as is expected since the associated NNs do not learn features. Conversely, we demonstrate that well-learned numerical metrics at finite-width exhibit an evolving metric-NTK, associated with feature learning. Our theory of NN metric flows therefore explains why NNs are better at learning Calabi–Yau metrics than fixed kernel methods, such as the Ricci flow. 

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4. 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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5. On the rotational symmetry of 3-dimensional κ-solutions

   https://doi.org/10.1515/crelle-2021-0037  

   Bamler, Richard H.; Kleiner, Bruce (August 2021, Journal für die reine und angewandte Mathematik)

   null (Ed.)

   Abstract In a recent paper, Brendle showed the uniqueness of the Bryant soliton among 3-dimensional κ-solutions.In this paper, we present an alternative proof for this fact and show that compact κ-solutions are rotationally symmetric.Our proof arose from independent work relating to our Strong Stability Theorem for singular Ricci flows. 

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