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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

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closed $\mapsto$ close $\:$ ? $\;\;\;\;$ –  Ricky Demer Feb 28 at 9:58
    
I am not an expert on the Ricci flow, but can't you use the existence of the flow for positive curvature metrics and corresponding estimates to deduce "how long" you need to flow, i.e., how close you are to the round sphere? –  Sebastian Feb 28 at 10:57
1  
For the case of an embedded surface in $\mathbb{R}^3$, I would try to adapt the proof of the constant curvature case; more precisely, one can hope to prove that the Gauss map, which is a diffeomorphism since curvature is positive, is in fact close to be an isometry. Maybe the Hilbert argument (see e.g. Montiel-Ros page 92 and below) can be adapted to get that the principal curvatures must be uniformly close one to another. –  Benoît Kloeckner Feb 28 at 19:38

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