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

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.