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

We introduce a new perspective on the classical Nirenberg problem of understanding possible Gauss curvatures metrics $S^{2}$ conformal to round metric. A key tool is employ smooth Cheeger–Gromov compactness theorem obtain general and essentially sharp priori estimates for $K$ contained in naturally defined stable regions. prove that such regions, map $u \to K_{g}$, $g = e^{2u}g_{+1}$ proper Fredholm with well-defined degree each component. This leads number existence non-existence results. also present proof generalization Moser even $S^{2}$.