scholarly journals Mean Curvature Flow of Spacelike Hypersurfaces near Null Initial Data

2003 ◽  
Vol 11 (2) ◽  
pp. 181-205 ◽  
Author(s):  
Klaus Ecker
Keyword(s):  
Initial Data ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Spacelike Hypersurfaces

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 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 convex entire graphs by curvature flows

2015 ◽  
Vol 1 (1) ◽  
Author(s):  
Roberta Alessandroni ◽  
Carlo Sinestrari
Keyword(s):  
Euclidean Space ◽  
Initial Data ◽  
Mean Curvature ◽  
Symmetric Function ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Principal Curvatures ◽  
The Mean ◽  
Entire Graphs

AbstractWe consider the evolution of an entire convex graph in euclidean space with speed given by a symmetric function of the principal curvatures. Under suitable assumptions on the speed and on the initial data, we prove that the solution exists for all times and it remains a graph. In addition, after appropriate rescaling, it converges to a homothetically expanding solution of the flow. In this way, we extend to a class of nonlinear speeds the well known results of Ecker and Huisken for the mean curvature flow.

scholarly journals Tan-concavity property for Lagrangian phase operators and applications to the tangent Lagrangian phase flow

2020 ◽  
pp. 2050116
Author(s):  
R. Takahashi
Keyword(s):  
Line Bundle ◽  
Initial Data ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Phase Flow ◽  
Phase Operator ◽  
Yang Mills ◽  
Positive Time ◽  
Phase Operators

We explore the tan-concavity of the Lagrangian phase operator for the study of the deformed Hermitian Yang–Mills (dHYM) metrics. This new property compensates for the lack of concavity of the Lagrangian phase operator as long as the metric is almost calibrated. As an application, we introduce the tangent Lagrangian phase flow (TLPF) on the space of almost calibrated [Formula: see text]-forms that fits into the GIT framework for dHYM metrics recently discovered by Collins–Yau. The TLPF has some special properties that are not seen for the line bundle mean curvature flow (i.e. the mirror of the Lagrangian mean curvature flow for graphs). We show that the TLPF starting from any initial data exists for all positive time. Moreover, we show that the TLPF converges smoothly to a dHYM metric assuming the existence of a [Formula: see text]-subsolution, which gives a new proof for the existence of dHYM metrics in the highest branch.

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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