# Mean convex embedded sphere and convex sphere

In $R^{n+1}$, suppose $M$ is smooth embedded hypersurface which is isomorphic to $S^{n}$. Is that the same when we talk about $M$ is mean convex (the normal vector, pointed inward), and $M$ is convex, ( in Huisken's sense）？

It's very different! Convex (in the Huisken sense) means that all the principal curvatures $\lambda_1,\ldots,\lambda_n$ are positive, i.e. $\lambda_1>0,\ldots,\lambda_n>0$. Mean convex only means that the mean curvature vector points inwards, i.e. just that the sum $\lambda_1+\ldots+\lambda_n$ is positive, which is obviously a much weaker curvature condition (e.g. $n-1$ principal curvatures could be negative as long as $\lambda_n$ is sufficiently positive).
Correspondingly the mean curvature flow behaves very differently. Huisken proved in 82 that convex hypersurfaces converge to a round point. On the other hand, in the mean convex case many interesting singularities can form. It's a deep theorem due to White that they are in fact all modeled on shrinking cylinders $S^j\times R^{n-j}$ (see also my paper with Kleiner for a more elementary proof). The neckpinch in $R^3$ described in Will Jagy's answer is the simplest example for such a singularity.