scholarly journals Ancient solution of mean curvature flow in space forms

Author(s):  
Li Lei ◽  
Hong-wei Xu ◽  
Entao Zhao
Author(s):  
Stephen Lynch ◽  
Huy The Nguyen

AbstractWe study solutions of high codimension mean curvature flow defined for all negative times, usually referred to as ancient solutions. We show that any compact ancient solution whose second fundamental form satisfies a certain natural pinching condition must be a family of shrinking spheres. Andrews and Baker (J Differ Geom 85(3):357–395, 2010) have shown that initial submanifolds satisfying this pinching condition, which generalises the notion of convexity, converge to round points under the flow. As an application, we use our result to simplify their proof.


2014 ◽  
Vol 25 (03) ◽  
pp. 1450021 ◽  
Author(s):  
Hongwei Xu ◽  
Yan Leng ◽  
Entao Zhao

The purpose of this paper is to investigate the convergence of the volume-preserving mean curvature flow of closed hypersurfaces in space forms. Assume that the initial hypersurface satisfies suitable integral curvature pinching conditions. We prove that along the volume-preserving mean curvature flow the initial hypersurface will be deformed to a totally umbilical sphere.


2020 ◽  
Vol 18 (1) ◽  
pp. 1518-1530
Author(s):  
Xuesen Qi ◽  
Ximin Liu

Abstract In this paper, we discuss the monotonicity of the first nonzero eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow (MCF). By imposing conditions associated with the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced MCF, and some special pinching conditions on the second fundamental form of the initial hypersurface, we prove that the first nonzero closed eigenvalues of the Laplace operator and the p-Laplace operator are monotonic under the forced MCF, respectively, which partially generalize Mao and Zhao’s work. Moreover, we give an example to specify applications of conclusions obtained above.


2017 ◽  
Vol 369 (12) ◽  
pp. 8319-8342 ◽  
Author(s):  
Glen Wheeler ◽  
Valentina-Mira Wheeler

Author(s):  
Peng Lu ◽  
Jiuru Zhou

AbstractWe construct the ancient solutions of the hypersurface flows in Euclidean spaces studied by B. Andrews in 1994.As time {t\rightarrow 0^{-}} the solutions collapse to a round point where 0 is the singular time. But as {t\rightarrow-\infty} the solutions become more and more oval. Near the center the appropriately-rescaled pointed Cheeger–Gromov limits are round cylinder solutions {S^{J}\times\mathbb{R}^{n-J}}, {1\leq J\leq n-1}. These results are the analog of the corresponding results in Ricci flow ({J=n-1}) and mean curvature flow.


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