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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. 

In this paper, we study a natural optimal control problem associated to the Paneitz obstacle problem on closed 4-dimensional Riemannian manifolds. We show the existence of an optimal control which is an optimal state and induces also a conformal metric with prescribed Q -curvature. We show also C ∞ -regularity of optimal controls and some compactness results for the optimal controls. In the case of the 4-dimensional standard sphere, we characterize all optimal controls. 

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. 

Abstract We show that the homogeneous and the 2-lobe Delaunay tori in the 3-sphere provide the only isothermic constrained Willmore tori in 3-space with Willmore energy below $$8\pi $$ 8 π . In particular, every constrained Willmore torus with Willmore energy below $$8\pi $$ 8 π and non-rectangular conformal class is non-degenerated.