Mean curvature evolution of entire graphs.(English)Zbl 0696.53036

This 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.
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
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