Local convexity and nonnegative curvature - Gromov's proof of the sphere theorem (Q1076988)

Following an idea of Gromov, the author gives a direct proof of the celebrated sphere theorem of Rauch-Berger-Klingenberg without appealing to injectivity radius estimates; in fact the usual injectivity radius estimate is another consequence of the proof. The proof is based on convexity arguments. An immersed hypersurface S in a Riemannian manifold M is called \(\epsilon\)-convex if all principal curvatures have the same sign and absolute value at least \(\epsilon\). With this definition the sphere theorem is a consequence of the following crucial result: Let \(M^ n\), \(n\geq 3\), be a complete Riemannian manifold with nonnegative curvature and S a closed connected (n-1)-manifold. Then for any \(\epsilon\)-convex immersion \(f: S\to M\) there is an immersion \(\hat f:\) \(D\to M\) where D is the standard n-disk, and a diffeomorphism \(\phi\) : \(\partial D\to S\) such that \(\hat f=f\circ \phi\) on \(\partial D\) and the mean curvature vector of f(S) is pointing towards \(\hat f(\)D). This is wrong for \(n=2\).