Deforming convex hypersurfaces by the square root of the scalar curvature. (English) Zbl 0608.53005

Chow, Bennett

Invent. Math. 87, 63-82 (1987).

Let \(F_ 0: S^ n\to {\mathbb{R}}^{n+1}\) be a smooth parametrization of a strictly convex hypersurface \(M_ 0\subset {\mathbb{R}}^{n+1}\), \(n\geq 2\), and consider the initial value problem \[ (*)\quad \partial F/\partial t (x,t)=-R^{1/2}(x,t)\cdot \nu (x,t) \]
\[ F(x,0)=F_ 0(x),\quad x\in S^ n. \] Here R denotes the scalar curvature and \(\nu\) the outward normal of \(M_ t\), the corresponding hypersurface at time t. Under the additional assumption that \(R(p)/H^ 2(p)>C(n)>0\) for all \(p\in M_ 0\), where H is the mean curvature and the constant C(n) is chosen so that the inequality will imply \(M_ 0\) is strictly convex, the author proves the following result: Problem (*) has a unique solution on a maximum time interval \(0\leq t<T\) and the \(M_ t's\) converge to a point as \(t\to T\). Moreover, the shape of the \(M_ t's\) approaches that of the standard round sphere. Similar results, with \(R^{1/2}\) replaced by H or the n-th root of the Gaussian curvature K, respectively, are due to G. Huisken [J. Differ. Geom. 20, 237-266 (1984; Zbl 0556.53001)] and the author [ibid. 22, 117-138 (1985; Zbl 0589.53005)].

Reviewer: R.Schneider

MSC:

53A07	Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
52A20	Convex sets in \(n\) dimensions (including convex hypersurfaces)
35G10	Initial value problems for linear higher-order PDEs
35K25	Higher-order parabolic equations

Keywords:

evolution equation; round sphere; convex hypersurface; scalar curvature

Citations:

Zbl 0556.53001; Zbl 0589.53005

References:

[1]	Cheng, S.Y., Yau, S.T.: Hypersurfaces with constant scalar curvature. Math. Ann.225, 195-204 (1977) · Zbl 0349.53041 · doi:10.1007/BF01425237
[2]	Chow, B.: Deforming convex hypersurfaces by then th root of the Gaussian curvature. J. Differ. Geom.22, 117-138 (1985) · Zbl 0589.53005
[3]	Gage, M., Hamilton, R.S.: The heat equation shrinking convex plane curves. J. Differ. Geom.23, 69-96 (1986) · Zbl 0621.53001
[4]	Hamilton, R.S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom.17, 255-306 (1982) · Zbl 0504.53034
[5]	Hamilton, R.S.: Four-manifolds with positive curvature operator. J. Differ. Geom. (in press) (1986) · Zbl 0628.53042
[6]	Hartman, P.: Hypersurfaces with nonnegative sectional curvatures and constantm th mean curvature. Trans. Am. Math. Soc.245, 363-374 (1978) · Zbl 0412.53027
[7]	Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom.20, 237-266 (1984) · Zbl 0556.53001
[8]	Huisken, G.: Ricci deformation of the metric on a Riemannian manifold. J. Differ. Geom. (in21, 47-62 (1985) · Zbl 0606.53026
[9]	Krylov, N.V., Safonov, M.V.: A certain property of solutions of parabolic equations with measurable coefficients. Math. USSR Izv.16, 151-164 (1981) · Zbl 0464.35035 · doi:10.1070/IM1981v016n01ABEH001283
[10]	Michael, J.H., Simon, L.M.: Sobolev and mean value inequalities on generalized submanifolds of ? n . Commun. Pure Appl. Math.26, 316-379 (1973) · Zbl 0252.53006 · doi:10.1002/cpa.3160260305
[11]	Stampacchia, G.: Equations elliptiques au second order à coéfficients discontinues. Sém. Math. Sup. 16, Les Presses de l’Université de Montreal, 1966
[12]	Trudinger, N.S.: Elliptic equations in nondivergence form. Proc. Miniconference on PDE, 1-16, Canberra 1981
[13]	Tso, Kaising: Deforming a hypersurface by its Gauss-Kronecker curvature. Commun. Pure Appl. Math.38, 867-882 (1985) · Zbl 0612.53005 · doi:10.1002/cpa.3160380615