quantitative version of the rigidity of the 2-sphere

Question

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

Answer 1

All hypersurfaces are $n$-dimensional.

Suppose $M$ and $N$ are two smooth, strictly convex hypersurfaces, such that $B_r(o)\subset M,N\subset B_R(o)$, where $B_x(y)$ denotes a ball centred at $x$ of radius $y.$ If $$|\frac{1}{K_M}-\frac{1}{K_L}|\leq \varepsilon,$$ then $d_H(M,N')\leq \gamma \varepsilon^{\frac{1}{n+1}}$. Here, $d_H$ denotes the Hausdorff distance and $N'$ is a suitable translation of $N,$ and $\gamma$ depends only on $n,r,R$ (see Schneider, ``CONVEX BODIES: THE BRUNN–MINKOWSKI THEORY"" Theorem 8.5.4).

Now suppose the Gauss curvature of the hypersurface $M$ satisfies $a\leq \frac{1}{K}\leq b$. First, We translate $M$ such that its centroid is at the origin $o.$ Second, by Lemma 3 of Cheng-Yau "On the Regularity of the Solution of the n-Dimensional Minkowski Problem", there are $r,R$ depending only on $a,b$ such that $$B_r(o)\subset M \subset B_R(o).$$Third, we may increase $R$ and decrease $r$ such that $$B_r(o)\subset B_{\sqrt[n]{\frac{a+b}{2}}}(o)\subset B_R(o).$$ Fourth, note that $$|\frac{1}{K}-\frac{a+b}{2}|\leq \frac{b-a}{2}.$$ Therefore, $$d_H(M',B_{\sqrt[n]{\frac{a+b}{2}}}(o))\leq \gamma \left(\frac{b-a}{2}\right)^\frac{1}{n+1},$$ where $M'$ is a translate of $M$ and $\gamma$ only depends on $n,a,b.$