scholarly journals Existence and regularity of mean curvature flow with transport term in higher dimensions

2015 ◽  
Vol 364 (3-4) ◽  
pp. 857-935 ◽  
Author(s):  
Keisuke Takasao ◽  
Yoshihiro Tonegawa
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Higher Dimensions ◽  
Curvature Flow ◽  
Transport Term

scholarly journals On the existence of mean curvature flow with transport term

2010 ◽  
pp. 251-277 ◽  
Author(s):  
Chun Liu ◽  
Norifumi Sato ◽  
Yoshihiro Tonegawa
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Transport Term

scholarly journals Stationary sets of the mean curvature flow with a forcing term

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Vesa Julin ◽  
Joonas Niinikoski
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Higher Dimensions ◽  
Curvature Flow ◽  
Planar Case ◽  
Forcing Term ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
The Mean ◽  
Flat Flow

Abstract We consider the flat flow solution to the mean curvature equation with forcing in ℝ n {\mathbb{R}^{n}} . Our main result states that tangential balls in ℝ n {\mathbb{R}^{n}} under a flat flow with a bounded forcing term will experience fattening, which generalizes the result in [N. Fusco, V. Julin and M. Morini, Stationary sets and asymptotic behavior of the mean curvature flow with forcing in the plane, preprint 2020, https://arxiv.org/abs/2004.07734] from the planar case to higher dimensions. Then, as in the planar case, we characterize stationary sets in ℝ n {\mathbb{R}^{n}} for a constant forcing term as finite unions of equisize balls with mutually positive distance.

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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