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1. Uniformization and the Yamabe Problem

   https://doi.org/10.1090/noti2967  

   McKeown, Stephen E; Ndiaye, Cheikh Birahim (August 2024, Notices of the American Mathematical Society)
2. Quantitative stability for minimizing Yamabe metrics

   https://doi.org/10.1090/btran/111  

   Engelstein, Max; Neumayer, Robin; Spolaror, Luca (May 2022, Transactions of the American Mathematical Society)

   On any closed Riemannian manifold of dimension , we prove that if a function nearly minimizes the Yamabe energy, then the corresponding conformal metric is close, in a quantitative sense, to a minimizing Yamabe metric in the conformal class. Generically, this distance is controlled quadratically by the Yamabe energy deficit. Finally, we produce an example for which this quadratic estimate is false. 

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3. Uniformization theorems: Between Yamabe and Paneitz

   https://doi.org/10.2140/pjm.2021.314.115  

   Ndiaye, Cheikh Birahim; Sire, Yannick; Sun, Liming (January 2021, Pacific Journal of Mathematics)
4. Kodaira dimension and the Yamabe problem, II

   https://doi.org/10.4310/CAG.2023.v31.n10.a3  

   Albanese, Michael; LeBrun, Claude (January 2023, Communications in Analysis and Geometry)

   Abstract. For compact complex surfaces (M^4,J) of Kaehler type, it was previously shown (LeBrun 1999) that the sign of the Yamabe invariant Y (M) only depends on the Kodaira dimension Kod(M,J). In this paper, we prove that this pattern in fact extends to all compact complex surfaces except those of class VII. In the process, we also reprove a result from (Albanese 2021) that explains why the exclusion of class VII is essential here. 

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5. The Yamabe flow on asymptotically flat manifolds

   https://doi.org/10.1515/crelle-2023-0052  

   Chen, Eric; Wang, Yi (September 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We study the Yamabe flow starting from an asymptotically flat manifold $(M^n,g_0)$(M^{n},g_{0}).We show that the flow converges to an asymptotically flat, scalar flat metric in a weighted global sense if $Y(M,\left[g_0\right])>0$Y(M,[g_{0}])>0, and show that the flow does not converge otherwise.If the scalar curvature is nonnegative and integrable, then the ADM mass at time infinity drops by the limit of the total scalar curvature along the flow. 

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