The Nirenberg problem of prescribed Gauss curvature on $S^2$

Abstract

We introduce a new perspective on the classical Nirenberg problem of understanding the possible Gauss curvatures of metrics on S^{2} conformal to the round metric. A key tool is to employ the smooth Cheeger–Gromov compactness theorem to obtain general and essentially sharp a priori estimates for Gauss curvatures K contained in naturally defined stable regions. We prove that in such stable regions, the map u \to K_{g} , g = e^{2u}g_{+1} is a proper Fredholm map with well-defined degree on each component. This leads to a number of new existence and non-existence results. We also present a new proof and generalization of the Moser theorem on Gauss curvatures of even conformal metrics on S^{2} . In contrast to previous work, the work here does not use any of the Sobolev-type inequalities of Trudinger–Moser–Aubin–Onofri.