Remark 1.1.2 (Mean curvature flow and normalization)4EDA6AEDEA07

Remark 1.1.2 (Mean curvature flow and normalization).4EDA6AEDEA07 For a smooth immersion $X:M^ n\times I\to \mathbb R^ N$, mean curvature flow is the geometric equation

where $\vec H$ is the trace of the second fundamental form and $\perp $ means normal projection. For hypersurfaces we use $H=\operatorname {div}_ M\nu $ and $\vec H=-H\nu $. Thus the outward normal of a round sphere has positive scalar mean curvature, and the sphere shrinks. For a smooth closed evolving submanifold,

Our self-shrinkers satisfy $\vec H=-X^\perp /2$ and generate $M_ t=\sqrt{-t}\, \Sigma $ for $t<0$. Self-expanders satisfy $\vec H=X^\perp /2$ and generate $M_ t=\sqrt t\, \Sigma $ for $t>0$. A translator satisfies $\vec H=V^\perp $ for a constant velocity $V$, with $M_ t=\Sigma +tV$. The translator pictures use upward unit speed unless stated otherwise. Dimensions, domains, scales, and time intervals are specified in each entry.