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 a priori estimates for Gauss curvatures $K$ which are in stable orbits of the conformal group $\mathrm{Conf}(S^{2})$. We prove that in such a stable region, the map $u \rightarrow K_{g}$, $g = e^{2u}g_{+1}$ 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.