scholarly journals Anisotropic mean curvature flow for two-dimensional surfaces in higher codimension: a numerical scheme

2008 ◽  
pp. 539-576 ◽  
Author(s):  
Paola Pozzi
Keyword(s):  
Mean Curvature ◽  
Numerical Scheme ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Two Dimensional ◽  
Anisotropic Mean Curvature

Skew mean curvature flow

2019 ◽  
Vol 21 (01) ◽  
pp. 1750090
Author(s):  
Chong Song ◽  
Jun Sun
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Embedding Theorem ◽  
Sobolev Type ◽  
Two Dimensional ◽  
Initial Value ◽  
Codimension Two ◽  
Surfaces In Euclidean Space ◽  
Short Time

The skew mean curvature flow (SMCF), which origins from the study of fluid dynamics, describes the evolution of a codimension two submanifold along its binormal direction. We study the basic properties of the SMCF and prove the existence of a short-time solution to the initial value problem of the SMCF of compact surfaces in Euclidean space [Formula: see text]. A Sobolev-type embedding theorem for the second fundamental forms of two-dimensional surfaces is also proved, which might be of independent interest.

scholarly journals Anisotropic mean curvature flow of Lipschitz graphs and convergence to self-similar solutions

2021 ◽  
Author(s):  
Annalisa Cesaroni ◽  
Heiko Kröner ◽  
Matteo Novaga
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Time Behavior ◽  
Long Time ◽  
Self Similar ◽  
The Stability ◽  
Lipschitz Graphs ◽  
Similar Solutions ◽  
Anisotropic Mean Curvature

We consider the anisotropic mean curvature flow of entire Lipschitz graphs. We prove existence and uniqueness of expanding self-similar solutions which are asymptotic to a prescribed cone, and we characterize the long time behavior of solutions, after suitable rescaling, when the initial datum is a sublinear perturbation of a cone. In the case of regular anisotropies, we prove the stability of self-similar solutions asymptotic to strictly mean convex cones, with respect to perturbations vanishing at infinity. We also show the stability of hyperplanes, with a proof which is novel also for the isotropic mean curvature flow.

scholarly journals APPROXIMATION OF THE ANISOTROPIC MEAN CURVATURE FLOW

2007 ◽  
Vol 17 (06) ◽  
pp. 833-844 ◽  
Author(s):  
ANTONIN CHAMBOLLE ◽  
MATTEO NOVAGA
Keyword(s):  
Mean Curvature ◽  
Variational Approach ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Mean Curvature Motion ◽  
Curvature Motion ◽  
Anisotropic Mean Curvature ◽  
Anisotropic Case

In this paper, we provide simple proofs of consistency for two well-known algorithms for mean curvature motion, Almgren–Taylor–Wang's1variational approach, and Merriman–Bence–Osher's algorithm.29Our techniques, based on the same notion of strict sub- and superflows, also work in the (smooth) anisotropic case.

Two-Dimensional Graphs Moving by Mean Curvature Flow

2002 ◽  
Vol 18 (2) ◽  
pp. 209-224 ◽  
Author(s):  
Jing Yi Chen ◽  
Jia Yu Li ◽  
Gang Tian
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Two Dimensional
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