Convexity estimates for flows by powers of the mean curvature.

*(English)*Zbl 1150.53024The study of the mean curvature flow (or, in general, of one of its powers) have attracted many mathematicians during the last 30 years. For example, G. Huisken showed [J. Differ. Geom. 20, 237–266 (1984; Zbl 0556.53001)] that convex surfaces remain convex under that flow and contract to a \` round\' point in finite time.

In the paper under review, the author continues the study initiated in [Math. Z. 251, No. 4, 721–733 (2005; Zbl 1087.53062)] where he investigated the evolution of a closed convex hypersurface in \({\mathbb R}^{n+1}\), \(F_0:M^n\to {\mathbb R}^{n+1}\), with positive mean curvature \(H\). There he showed that there exists a unique smooth solution \(F(\cdot,t):M^n\times[0,T)\to {\mathbb R}^{n+1}\) to the initial value problem given by the \(H^k\)-flow on a maximal finite interval \([0,T)\): \[ \begin{aligned} F(\cdot,0) &=F_0(\cdot)\\ \frac{dF}{dt}(\cdot,t) &=-H^k(\cdot,t)\nu(\cdot,t) \end{aligned} \] where \(k>0\) and \(\nu\) is the outer unit normal such that \(-H\nu={\mathbf H}\) is the mean curvature vector.

In this paper the author presents an extension of the previous result by using that by the arithmetic-geometric mean inequality we have \(0\leq K/H^n\leq 1/n^n\), with equality on the right side if and only if all eigenvalues are equal. The main result of the paper states as follows.

“For \(k\geq 1\) there exists a nonnegative constant \(C(n,k)<1/n^n\) such that the following holds: If the initial hypersurface is pinched in the sense that \[ \frac{K(p)}{H^n(p)}>C(n,k), \] for all \(p\in M\), then this is preserved under the \(H^k\)-flow. The constant \(C(n,k)\) is increasing in \(k\), \(\lim_{k\to 1}C(n,k)=0\) and \(\lim_{k\to \infty}C(n,k)=1/n^n\). Furthermore, the rescaled embeddings \[ \tilde F(\tau,p):=((k+1)n^k(T-t))^{-1/(k+1)}(F(\tau,p)-x_0) \] converge for \(\tau\to\infty\) exponentially in the \(C^\infty\)-topology to the unit sphere. Here \(\tau:=-(k+1)^{-1}n^{-k}\log(1-t/T)\), where \(T\) is the maximal time of existence of the unrescaled flow and \(x_0\) is the point in \({\mathbb R}^{n+1}\) where the surfaces contract to”.

The paper also contains an appendix, written jointly with O. Schnürer, where they give an extension for surfaces in the 3-dimensional Euclidean space. They show that for \(1\leq k\leq 5\) no initial pinching condition is needed to ensure that the rescaled embeddings converge to the unit sphere.

In the paper under review, the author continues the study initiated in [Math. Z. 251, No. 4, 721–733 (2005; Zbl 1087.53062)] where he investigated the evolution of a closed convex hypersurface in \({\mathbb R}^{n+1}\), \(F_0:M^n\to {\mathbb R}^{n+1}\), with positive mean curvature \(H\). There he showed that there exists a unique smooth solution \(F(\cdot,t):M^n\times[0,T)\to {\mathbb R}^{n+1}\) to the initial value problem given by the \(H^k\)-flow on a maximal finite interval \([0,T)\): \[ \begin{aligned} F(\cdot,0) &=F_0(\cdot)\\ \frac{dF}{dt}(\cdot,t) &=-H^k(\cdot,t)\nu(\cdot,t) \end{aligned} \] where \(k>0\) and \(\nu\) is the outer unit normal such that \(-H\nu={\mathbf H}\) is the mean curvature vector.

In this paper the author presents an extension of the previous result by using that by the arithmetic-geometric mean inequality we have \(0\leq K/H^n\leq 1/n^n\), with equality on the right side if and only if all eigenvalues are equal. The main result of the paper states as follows.

“For \(k\geq 1\) there exists a nonnegative constant \(C(n,k)<1/n^n\) such that the following holds: If the initial hypersurface is pinched in the sense that \[ \frac{K(p)}{H^n(p)}>C(n,k), \] for all \(p\in M\), then this is preserved under the \(H^k\)-flow. The constant \(C(n,k)\) is increasing in \(k\), \(\lim_{k\to 1}C(n,k)=0\) and \(\lim_{k\to \infty}C(n,k)=1/n^n\). Furthermore, the rescaled embeddings \[ \tilde F(\tau,p):=((k+1)n^k(T-t))^{-1/(k+1)}(F(\tau,p)-x_0) \] converge for \(\tau\to\infty\) exponentially in the \(C^\infty\)-topology to the unit sphere. Here \(\tau:=-(k+1)^{-1}n^{-k}\log(1-t/T)\), where \(T\) is the maximal time of existence of the unrescaled flow and \(x_0\) is the point in \({\mathbb R}^{n+1}\) where the surfaces contract to”.

The paper also contains an appendix, written jointly with O. Schnürer, where they give an extension for surfaces in the 3-dimensional Euclidean space. They show that for \(1\leq k\leq 5\) no initial pinching condition is needed to ensure that the rescaled embeddings converge to the unit sphere.

Reviewer: Pascual Lucas Saorín (Murcia)

##### MSC:

53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |

35B40 | Asymptotic behavior of solutions to PDEs |

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\textit{F. Schulze}, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 5, No. 2, 261--277 (2006; Zbl 1150.53024)

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##### References:

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[9] | F. Schulze, Evolution of convex hypersurfaces by powers of the mean curvature, Math. Z. 251 (2005), 721-733. Zbl1087.53062 MR2190140 · Zbl 1087.53062 · doi:10.1007/s00209-004-0721-5 |

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