Surfaces of revolution with monotonic increasing curvature and an application to the equation $\Delta u=1-K e^{2u}$ on $S^{2}$

The geometric result that a compact surface of revolution in cannot have monotonic increasing curvature is proved and applied to show that the equation <!-- MATH $\Delta u = 1 - K{e^{2u}}$ --> , on , has no axially symmetric solutions u, given axially symmetric data K.