When maximum-principle functions cease to exist

We consider geometric flow equations for contracting and expanding normal velocities, including powers of the Gauss curvature, of the mean curvature, and of the norm of the second fundamental form, and ask whether - after appropriate rescaling - closed strictly convex surfaces converge to spheres. To prove this, many authors use certain functions of the principal curvatures, which we call maximum-principle functions. We show when such functions cease to exist and exist, while presenting newly discovered maximum-principle functions.