The k-Yamabe problem

The Yamabe Problem

Existence theorems of the fractional Yamabe problem

Let X be an asymptotically hyperbolic manifold and M its conformal infinity. This paper is devoted to deduce several existence results of the fractional Yamabe problem on M under various geometric assumptions on X and M: Firstly, we handle when the boundary M has a point at which the mean curvature is negative. Secondly, we re-encounter the case when M has zero mean curvature and is either non-...

Kodaira Dimension and the Yamabe Problem

The Yamabe invariant Y (M) of a smooth compact manifold is roughly the supremum of the scalar curvatures of unit-volume constant-scalar curvature Riemannian metrics g on M . (To be absolutely precise, one only considers constant-scalar-curvature metrics which are Yamabe minimizers, but this does not affect the sign of the answer.) If M is the underlying smooth 4-manifold of a complex algebraic ...

A Compactness Theorem for the Yamabe Problem

In this paper, we prove compactness for the full set of solutions to the Yamabe Problem if n ≤ 24. After proving sharp pointwise estimates at a blowup point, we prove the Weyl Vanishing Theorem in those dimensions, and reduce the compactness question to showing positivity of a quadratic form. We also show that this quadratic form has negative eigenvalues if n ≥ 25.

Compactness of solutions to the Yamabe problem

We establish compactness of solutions to the Yamabe problem on any smooth compact connected Riemannian manifold (not conformally diffeomorphic to standard spheres) of dimension n 7 as well as on any manifold of dimension n 8 under some additional hypothesis. To cite this article: Y.Y. Li, L. Zhang, C. R. Acad. Sci. Paris, Ser. I 338 (2004).  2004 Académie des sciences. Published by Elsevier SA...