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Let $M^n$ be a compact, connected, orientable hypersurface of the unit sphere $S^{n+1} \subset \mathbb{R}^{n+2}$. Suppose $M$ is contained in the northern hemisphere $S_+^{n+1}$ and has nonzero principal curvatures everywhere, i.e., has nonzero gaussian curvature everywhere. It seems to me that if we compress $M$ somehow in the direction of the north pole, its principal curvatures will, in absolute value, get arbitrarily large.

Is it true? If so, is there a way to show it without explicitly exhibiting a map $S_+^{n+1} \to S_+^{n+1}$ that does the job? Instead of compressing $M$ towards the north pole, let us think more generally; is it true that there exists a diffeomorphic copy of $M$ in $S_+^{n+1}$ having prinicipal curvatures say, in absolute value bigger than 1?

Thanks for your help and thoughts!

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    $\begingroup$ You can write out the Ptolemaic coordinates on the sphere with an explicit expression for the Riemannian metric, so that dilation in those coordinates gives a very explicit compression toward the north pole. You can see explicitly an orthonormal frame and how it transforms in those coordinates, I think, so you should be able to see how shape operators transform. $\endgroup$
    – Ben McKay
    Commented Apr 15, 2016 at 20:48
  • $\begingroup$ @BenMcKay What are Ptolemaic coordinates? $\endgroup$
    – Eduardo Longa
    Commented Apr 15, 2016 at 23:25
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    $\begingroup$ I am not sure which one Ben McKay means specifically, but you can do his construction in either stereographic projection or orthographic projection. $\endgroup$
    – Willie Wong
    Commented Apr 29, 2016 at 19:06

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