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1. Blow-up sets of Ricci curvatures of complete conformal metrics

   https://doi.org/10.1515/crelle-2025-0020  

   Han, Qing; Shen, Weiming; Wang, Yue (March 2025, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract A version of the singular Yamabe problem in smooth domains in a closed manifoldyields complete conformal metrics with negative constant scalar curvatures.In this paper, we study the blow-up phenomena of Ricci curvatures of these metrics on domains whose boundary is close to a certain limit set of a lower dimension.We will characterize the blow-up set according to the Yamabe invariant of the underlying manifold.In particular, we will prove that all points in the lower dimension part of the limit set belong to the blow-up set on manifolds not conformally equivalent to the standard sphere and that all but one point in the lower dimension part of the limit set belong to the blow-up set on manifolds conformally equivalent to the standard sphere.In certain cases, the blow-up set can be the entire manifold.We will demonstrate by examples that these results are optimal. 

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2. 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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3. 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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4. Fractional Yamabe Problem on Locally Flat Conformal Infinities of Poincaré-Einstein Manifolds

   https://doi.org/10.1093/imrn/rnad195  

   Mayer, Martin; Ndiaye, Cheikh_Birahim (August 2023, International Mathematics Research Notices)

   Abstract We study the fractional Yamabe problem first considered by Gonzalez-Qing [36] on the conformal infinity $$(M^{n}, \;[h])$$ of a Poincaré-Einstein manifold $$(X^{n+1}, \;g^{+})$$ with either $n=2$ or $$n\geq 3$$ and $$(M^{n}, \;[h])$$ locally flat, namely $(M, h),$ is locally conformally flat. However, as for the classical Yamabe problem, because of the involved quantization phenomena, the variational analysis of the fractional one exhibits a local situation and also a global one. The latter global situation includes the case of conformal infinities of Poincaré-Einstein manifolds of dimension either $n=2$ or of dimension $$n\geq 3$$ and which are locally flat, and hence the minimizing technique of Aubin [4] and Schoen [48] in that case clearly requires an analogue of the positive mass theorem of Schoen-Yau [49], which is not known to hold. Using the algebraic topological argument of Bahri-Coron [8], we bypass the latter positive mass issue and show that any conformal infinity of a Poincaré-Einstein manifold of dimension either $n=2$ or of dimension $$n\geq 3$$ and which is locally flat admits a Riemannian metric of constant fractional scalar curvature. 

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5. Asymptotics of the Poisson kernel and Green's functions of the fractional conformal Laplacian

   https://doi.org/10.3934/dcds.2022085  

   Mayer, Martin; Ndiaye, Cheikh Birahim (January 2022, Discrete and Continuous Dynamical Systems)

   We study the asymptotics of the Poisson kernel and Green's functions of the fractional conformal Laplacian for conformal infinities of asymptotically hyperbolic manifolds. We derive sharp expansions of the Poisson kernel and Green's functions of the conformal Laplacian near their singularities. Our expansions of the Green's functions answer the first part of the conjecture of Kim-Musso-Wei[21] in the case of locally flat conformal infinities of Poincare-Einstein manifolds and together with the Poisson kernel asymptotic is used also in our paper [25] to show solvability of the fractional Yamabe problem in that case. Our asymptotics of the Green's functions on the general case of conformal infinities of asymptotically hyperbolic space is used also in [29] to show solvability of the fractional Yamabe problem for conformal infinities of dimension \begin{document}$ 3 $$\end{document} and fractional parameter in \begin{document}$$ (\frac{1}{2}, 1) $$\end{document}$ corresponding to a global case left by previous works. 

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