I am looking for a quantitaive version of the following theorem:
A compact surface with $K\equiv 1$ is isometric to the round sphere.
Of course I get the Berger, Brendle-Schoen Theorem which insures that if we are $1/4$ pinching, the surface is diffeomorphic to the sphere.
I also know the result of de-Lellis and Muller which asserts that if the $L^2$-norm of the trace-less second fundamental form is small enough, the surface is close (as an immersed object) to the round sphere.
Unfortunately it does not seem to imply the following:
if $max K/min K$ is small enough, we are close to a round sphere.
Does any one know something in that spirit? Thanks