scholarly journals On mean curvature flow of spacelike hypersurfaces in asymptotically flat spacetimes

1993 ◽  
Vol 55 (1) ◽  
pp. 41-59 ◽  
Author(s):  
Klaus Ecker
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
A Priori Estimates ◽  
A Priori ◽  
Curvature Flow ◽  
Lorentzian Manifold ◽  
Asymptotically Flat ◽  
Spacelike Hypersurfaces ◽  
Flat Spacetimes ◽  
Weaker Conditions

AbstractWe prove a priori estimates for the gradient and curvature of spacelike hypersurfaces moving by mean curvature in a Lorentzian manifold. These estimates are obtained under much weaker conditions than have been previously assumed. We also use mean curvature flow in the construction of maximal slices in asymptotically flat spacetimes. An essential tool is a maximum principle for sub-solutions of a parabolic operator on complete Riemannian manifolds with time-dependent metric.

scholarly journals Mean Curvature Flow in Asymptotically Flat Product Spacetimes

2020 ◽  
Author(s):  
Klaus Kröncke ◽  
Oliver Lindblad Petersen ◽  
Felix Lubbe ◽  
Tobias Marxen ◽  
Wolfgang Maurer ◽  
...  
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Asymptotically Flat

Lower semicontinuity of mass under C0{C^{0}} convergence and Huisken’s isoperimetric mass

2019 ◽  
Vol 2019 (756) ◽  
pp. 227-257 ◽  
Author(s):  
Jeffrey L. Jauregui ◽  
Dan A. Lee
Keyword(s):  
Scalar Curvature ◽  
Lower Semicontinuity ◽  
Mean Curvature ◽  
Total Mass ◽  
Mean Curvature Flow ◽  
Lower Semicontinuous ◽  
Curvature Flow ◽  
Three Dimensions ◽  
Asymptotically Flat ◽  
Limit Space

AbstractGiven a sequence of asymptotically flat 3-manifolds of nonnegative scalar curvature with outermost minimal boundary, converging in the pointed {C^{0}} Cheeger–Gromov sense to an asymptotically flat limit space, we show that the total mass of the limit is bounded above by the liminf of the total masses of the sequence. In other words, total mass is lower semicontinuous under such convergence. In order to prove this, we use Huisken’s isoperimetric mass concept, together with a modified weak mean curvature flow argument. We include a brief discussion of Huisken’s work before explaining our extension of that work. The results are all specific to three dimensions.

scholarly journals Contracting spacelike hypersurfaces by their inverse mean curvature

2000 ◽  
Vol 68 (3) ◽  
pp. 285-300 ◽  
Author(s):  
Michael Holder
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Parabolic System ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Nonlinear Parabolic System ◽  
Inverse Mean Curvature Flow ◽  
Nonlinear Parabolic ◽  
Spacelike Hypersurfaces

AbstractIn this work we study the behaviour of compact, smooth, orientable, spacelike hypersurfaces without boundary, which are immersed in cosmological spacetimes and move under the inverse mean curvature flow. We prove longtime existence and regularity of a solution to the corresponding nonlinear parabolic system of partial differential equations.

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.

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