an:03886661
Zbl 0556.53001
Huisken, Gerhard
Flow by mean curvature of convex surfaces into spheres
EN
J. Differ. Geom. 20, 237-266 (1984).
00149187
1984
j
53A07 35G10 35K25 52A20
convex hypersurface; mean curvature; evolution equation
Let \(M_ 0\) be a smooth closed convex hypersurface with everywhere positive curvatures in Euclidean space \({\mathbb{R}}^{n+1}\). Suppose that \(M_ 0\) is smoothly deformed (that is, embedded into a smooth family \((M_ t\), \(t\geq 0)\) of hypersurfaces depending on a time parameter) such that, at every time t, the rate of change of \(M_ t\) in direction of the inward unit normal vector is equal to the (positive) mean curvature of the hypersurface at the point considered. The author shows that the parabolic evolution equation describing the problem has a smooth solution on a finite time interval \(0\leq t\leq T\), and the \(M_ t's\) converge to a point as \(t\to T\). Moreover, if the surfaces undergo suitable homotheties and the time parameter is transformed appropriately into a parameter \(\tilde t,\) \(0\leq \tilde t<\infty\), it is shown that the normalized surfaces converge to a sphere in the \(C^{\infty}\)-topology as \(\tilde t\to \infty\). As the author says, his approach is inspired by a paper of \textit{R. S. Hamilton} [J. Differ. Geom. 17, 255-306 (1982; Zbl 0504.53034)], and he can use many of the methods developed there. The case \(n=1\) was treated by \textit{M. E. Gage} [Invent. Math. 76, 357-364 (1984; Zbl 0542.53004)].
\{Reviewer's remark: A similar problem, related in spirit though not in the methods, was treated by \textit{W. J. Firey} [Mathematika 21, 1-11 (1974; Zbl 0311.52003)]. There the rate of change of the support function is proportional to the Gauss curvature.\}
R.Schneider
Zbl 0504.53034; Zbl 0542.53004; Zbl 0311.52003