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1. Bergman metrics with constant holomorphic sectional curvatures

   https://doi.org/10.1515/crelle-2025-0013  

   Huang, Xiaojun; Li, Song-Ying; Treuer, John N (March 2025, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract The paper studies complex manifolds whose Bergman metrics are incomplete but have constant holomorphic sectional curvature.We will construct a real analytic unbounded domain in ${\mathbb{C}}^2$\mathbb{C}^{2}whose Bergman metric is well-defined and has a positive constant holomorphic sectional curvature, which appears to be the first example of this kind.We will answer a long standing folklore conjecture that a Stein manifold has a negative constant holomorphic sectional curvature if and only if it is biholomorphic to a ball with a pluripolar set removed.Together with the uniqueness of a moment problem in the appendix of the paper provided by John Treuer, we will show that, under natural assumptions, there is no complex manifold whose Bergman metric is flat. 

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2. Ricci flow does not preserve positive sectional curvature in dimension four

   https://doi.org/10.1007/s00526-022-02335-z  

   Bettiol, Renato G.; Krishnan, Anusha M. (November 2022, Calculus of Variations and Partial Differential Equations)

   Abstract We find examples of cohomogeneity one metrics on$$S^4$$ $S^4$and$$\mathbb {C}P^2$$ $C{P}^2$with positive sectional curvature that lose this property when evolved via Ricci flow. These metrics are arbitrarily small perturbations of Grove–Ziller metrics with flat planes that become instantly negatively curved under Ricci flow. 

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3. 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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4. Harmonic branched coverings and uniformization of CAT( k ) spheres

   https://doi.org/10.1515/crelle-2021-0031  

   Breiner, Christine; Mese, Chikako (August 2021, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let S be a surface with a metric d satisfying an upper curvature bound in the sense of Alexandrov (i.e. via triangle comparison).We show that an almost conformal harmonic map from a surface into ( S , d ) {(S,d)} is a branched covering. As a consequence, if ( S , d ) {(S,d)} is homeomorphically equivalent to the 2-sphere 𝕊 2 {\mathbb{S}^{2}} , then it is conformally equivalent to 𝕊 2 {\mathbb{S}^{2}} . 

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5. Four-manifolds of pinched sectional curvature

   https://doi.org/10.1016/j.aim.2018.07.003  

   Cao, Xiaodong; Tran, Hung (January 2022, Pacific journal of mathematics)

   In this paper, we study closed four-dimensional manifolds. In particular, we show that under various pinching curvature conditions (for example, the sectional curvature is no more than 5 6 of the smallest Ricci eigenvalue), the manifold is definite. If restricting to a metric with harmonic Weyl tensor, then it must be self-dual or anti-self-dual under the same conditions. Similarly, if restricting to an Einstein metric, then it must be either the complex projective space with its Fubini-Study metric, the round sphere, or their quotients. Furthermore, we also classify Einstein manifolds with positive intersection form and an upper bound on the sectional curvature. 

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