scholarly journals Mean curvature flow with surgeries of two–convex hypersurfaces

2008 ◽  
Vol 175 (1) ◽  
pp. 137-221 ◽  
Author(s):  
Gerhard Huisken ◽  
Carlo Sinestrari
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Convex Hypersurfaces

scholarly journals No mass drop for mean curvature flow of mean convex hypersurfaces

2008 ◽  
Vol 142 (2) ◽  
pp. 283-312 ◽  
Author(s):  
Jan Metzger ◽  
Felix Schulze
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Mean Convex ◽  
Convex Hypersurfaces

scholarly journals Mean Curvature Flow of Mean Convex Hypersurfaces

2016 ◽  
Vol 70 (3) ◽  
pp. 511-546 ◽  
Author(s):  
Robert Haslhofer ◽  
Bruce Kleiner
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Mean Convex ◽  
Convex Hypersurfaces

scholarly journals A survey on Inverse mean curvature flow in ROSSes

2017 ◽  
Vol 4 (1) ◽  
pp. 245-262
Author(s):  
Giuseppe Pipoli
Keyword(s):  
Mean Curvature ◽  
Symmetric Spaces ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Unified Approach ◽  
Inverse Mean Curvature Flow ◽  
Rank One ◽  
Similarities And Differences ◽  
Mean Convex ◽  
Convex Hypersurfaces

AbstractIn this survey we discuss the evolution by inverse mean curvature flow of star-shaped mean convex hypersurfaces in non-compact rank one symmetric spaces. We show similarities and differences between the case considered, with particular attention to how the geometry of the ambient manifolds influences the behaviour of the evolution. Moreover we try, when possible, to give an unified approach to the results present in literature.

scholarly journals Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow

2020 ◽  
Vol 18 (1) ◽  
pp. 1518-1530
Author(s):  
Xuesen Qi ◽  
Ximin Liu
Keyword(s):  
Laplace Operator ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Second Fundamental Form ◽  
Coefficient Function ◽  
First Eigenvalue ◽  
Initial Hypersurface ◽  
The Mean ◽  
The Laplace Operator

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.

scholarly journals Mean curvature flow with free boundary outside a hypersphere

2017 ◽  
Vol 369 (12) ◽  
pp. 8319-8342 ◽  
Author(s):  
Glen Wheeler ◽  
Valentina-Mira Wheeler
Keyword(s):  
Free Boundary ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow

scholarly journals Ancient solutions for Andrews’ hypersurface flow

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Peng Lu ◽  
Jiuru Zhou
Keyword(s):  
Ricci Flow ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Euclidean Spaces ◽  
Round Cylinder ◽  
Ancient Solutions ◽  
Singular Time

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.

scholarly journals Noncollapsing in mean-convex mean curvature flow

2012 ◽  
Vol 16 (3) ◽  
pp. 1413-1418 ◽  
Author(s):  
Ben Andrews
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Mean Convex
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