Geometry of complete hypersurfaces evolved by mean curvature flow (Q1812244)

Let \(M\) be a complete \(n\)-dimensional manifold without boundary and let \(F_t: M^n\to\mathbb R^{n+1}\) be a one-parameter family of smooth hypersurface immersions in Euclidean space. We say that \(M_t = F_t(M^n)\) is a solution of the mean curvature flow (MCF) problem if \(F_t\) satisfies \[ \left\{\begin{aligned} \frac{\partial}{\partial t}F(X,t) &= -H(X, t)N(X,t),\quad X\in M,\, t\geq 0,\\ F(\cdot,0) &= F_0(\cdot), \end{aligned}\right.\tag{1} \] where \(H(X,t)\) and \(N(X,t)\) are the mean curvature and the unit normal vector field, respectively, and \(F_0\) describes the immersion of some given initial hypersurface. The author is interested in the geometric behaviour of solutions of the problem (1) before the singularities occur. The attention is confined to solutions which are smooth and compact, or smooth and complete with bounded second fundamental form. It is shown how the distance changes between two points under the MCF. The estimates of the derivatives of the second fundamental form from the bound of the second fundamental form are obtained. It is useful in discussing the existence of the long time solution and the classification of singularities. The author obtains a maximum principle for mean curvature flow which is more convenient than that used by K.~Ecker and G.~Huisken. The curvature of complete manifolds at infinity is studied, and the volume of the complete weakly convex solution to the MCF is discussed.