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 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!

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!