A compactness theorem for the yamabe problem

Abstract

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. © 2009 Applied Probability Trust.