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

  • $\begingroup$ closed $\mapsto$ close $\:$ ? $\;\;\;\;$ $\endgroup$ – user5810 Feb 28 '14 at 9:58
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    $\begingroup$ 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? $\endgroup$ – Sebastian Feb 28 '14 at 10:57
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    $\begingroup$ 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. $\endgroup$ – Benoît Kloeckner Feb 28 '14 at 19:38
  • $\begingroup$ If I had to guess, the easiest way to go is to use the fact that any metric on the $2$-sphere is conformal to the standard one and use the PDE corresponding to the formula for the Gauss curvature of the pinched metric in terms of the conformal factor and the Laplacian for the standard metric. $\endgroup$ – Deane Yang Oct 18 '15 at 21:14

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 $n,a,b.$


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