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 Fre...

A Bonnesen-type inequality is a sharp isoperimetric inequality that includes an error estimate in terms of inscribed and circumscribed regions. A kinematic technique is used to prove a Bonnesen-type inequality for the Euclidean sphere (having constant Gauss curvature κ > 0) and the hyperbolic plane (having constant Gauss curvature κ < 0). These generalized inequalities each converge to the clas...