Mean curvature evolution of entire graphs.

*(English)*Zbl 0696.53036This paper deals with the motion of a surface by its mean curvature vector. The context here is a one parameter family \(F_ i=F(,t)\) with each \(F_ i\) a smooth immersion into \({\mathbb{R}}^{n+1}\) of the fixed n- dimensional manifold M. Motion by the mean curvature vector is described by the differential equation
\[
(*)\quad (d/dt)F(p,t)=H(p,t),
\]
where H(p,t) is the mean curvature vector of the hypersurface \(M_ t=F_ t(M)\) at F(p,t). Typically, initial data \(F(p,0)=F_ 0(p)\) is also imposed. A further condition imposed in this paper is that \(M=M_ 0\) be an entire graph, that is, that there exists a unit vector \(\omega\) such that, for one choice of unit normal vector \(\nu\) on M, \(<\nu,\omega >>0\) holds everywhere and the orthogonal projection from M to a hyperplane determined by \(\omega\) is surjective.

One of the main results is that if there exists \(\epsilon >0\) such that \(<\nu,\omega >\geq \epsilon\) holds everywhere on \(M_ 0\), then there is a solution of the initial value problem which is smooth for all \(t\geq 0\). The proof relies on a priori estimates on the hypersurfaces. Another main result concerns the asymptotic behavior of such a long-time solution: If some additional initial estimates hold, then after appropriate rescaling, the surfaces converge to a solution of the equation \(F^{\perp}=H\) characteristic of expanding self-similar solutions of (*). An example is given to show the optimality of the latter result, and in an appendix it is shown that the only entire graphs which satisfy \(F^{\perp}=-H\) [the equation corresponding to contracting self-similar solutions of (*)] are, in fact, planes.

One of the main results is that if there exists \(\epsilon >0\) such that \(<\nu,\omega >\geq \epsilon\) holds everywhere on \(M_ 0\), then there is a solution of the initial value problem which is smooth for all \(t\geq 0\). The proof relies on a priori estimates on the hypersurfaces. Another main result concerns the asymptotic behavior of such a long-time solution: If some additional initial estimates hold, then after appropriate rescaling, the surfaces converge to a solution of the equation \(F^{\perp}=H\) characteristic of expanding self-similar solutions of (*). An example is given to show the optimality of the latter result, and in an appendix it is shown that the only entire graphs which satisfy \(F^{\perp}=-H\) [the equation corresponding to contracting self-similar solutions of (*)] are, in fact, planes.

Reviewer: H.Parks

##### MSC:

53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |

53A07 | Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces |