Compactness of solutions to the Yamabe problem . III

2 00 6 Compactness of solutions to the Yamabe problem . III YanYan

For a sequence of blow up solutions of the Yamabe equation on non-locally confonformally flat compact Riemannian manifolds of dimension 10 or 11, we establish sharp estimates on its asymptotic profile near blow up points as well as sharp decay estimates of the Weyl tensor and its covariant derivatives at blow up points. If the Positive Mass Theorem held in dimensions 10 and 11, these estimates ...

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.

A compactness theorem for a fully nonlinear Yamabe problem under a lower Ricci curvature bound

We prove compactness of solutions of a fully nonlinear Yamabe problem satisfying a lower Ricci curvature bound, when the manifold is not conformally diffeomorphic to the standard sphere. This allows us to prove the existence of solutions when the associated cone Γ satisfies μ+Γ ≤ 1, which includes the σk−Yamabe problem for k not smaller than half of the dimension of the manifold.

A fully nonlinear version of the Yamabe problem on locally conformally flat manifolds with umbilic boundary

We prove existence and compactness of solutions to a fully nonlinear Yamabe problem on locally conformally flat Riemannian manifolds with umbilic boundary.

1 N ov 2 00 4 Compactness of solutions to the Yamabe problem . II YanYan