scholarly journals Long-time existence of mean curvature flow with external force fields

2008 ◽  
Vol 234 (2) ◽  
pp. 311-324 ◽  
Author(s):  
Huai-Yu Jian ◽  
Yannan Liu
Keyword(s):  
External Force ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Force Fields ◽  
Curvature Flow ◽  
Long Time Existence ◽  
Long Time ◽  
Time Existence

scholarly journals Minkowski inequalities and constrained inverse curvature flows in warped spaces

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Julian Scheuer
Keyword(s):  
Mean Curvature Flow ◽  
Broad Class ◽  
Higher Dimensions ◽  
Curvature Flow ◽  
Radial Coordinate ◽  
Inverse Mean Curvature Flow ◽  
Long Time Existence ◽  
Long Time ◽  
Smooth Convergence ◽  
Time Existence

AbstractThis paper deals with locally constrained inverse curvature flows in a broad class of Riemannian warped spaces. For a certain class of such flows, we prove long-time existence and smooth convergence to a radial coordinate slice. In the case of two-dimensional surfaces and a suitable speed, these flows enjoy two monotone quantities. In such cases, new Minkowski type inequalities are the consequence. In higher dimensions, we use the inverse mean curvature flow to obtain new Minkowski inequalities when the ambient radial Ricci curvature is constantly negative.

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 A flow method for a generalization of $ L_{p} $ Christofell-Minkowski problem

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Boya Li ◽  
Hongjie Ju ◽  
Yannan Liu
Keyword(s):  
Initial Data ◽  
Curvature Flow ◽  
Minkowski Problem ◽  
Smooth Solutions ◽  
Flow Method ◽  
Long Time Existence ◽  
Long Time ◽  
Anisotropic Curvature ◽  
Time Existence

<p style='text-indent:20px;'>In this paper, a generalitzation of the <inline-formula><tex-math id="M2">\begin{document}$ L_{p} $\end{document}</tex-math></inline-formula>-Christoffel-Minkowski problem is studied. We consider an anisotropic curvature flow and derive the long-time existence of the flow. Then under some initial data, we obtain the existence of smooth solutions to this problem for <inline-formula><tex-math id="M3">\begin{document}$ c = 1 $\end{document}</tex-math></inline-formula>.</p>

scholarly journals A local regularity theorem for mean curvature flow with triple edges

2020 ◽  
Vol 2020 (758) ◽  
pp. 281-305 ◽  
Author(s):  
Felix Schulze ◽  
Brian White
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Natural Extension ◽  
Regularity Result ◽  
Curvature Flow ◽  
Higher Order ◽  
Local Regularity ◽  
Initial Surface ◽  
Short Time ◽  
Time Existence

AbstractMean curvature flow of clusters of n-dimensional surfaces in {\mathbb{R}^{n+k}} that meet in triples at equal angles along smooth edges and higher order junctions on lower-dimensional faces is a natural extension of classical mean curvature flow. We call such a flow a mean curvature flow with triple edges. We show that if a smooth mean curvature flow with triple edges is weakly close to a static union of three n-dimensional unit density half-planes, then it is smoothly close. Extending the regularity result to a class of integral Brakke flows, we show that this implies smooth short-time existence of the flow starting from an initial surface cluster that has triple edges, but no higher order junctions.

scholarly journals Deforming metrics of foliations

2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Vladimir Rovenski ◽  
Robert Wolak
Keyword(s):  
Heat Flow ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Existence Results ◽  
Long Time Existence ◽  
Long Time ◽  
Main Observation ◽  
The Mean ◽  
Conformal Flow ◽  
Time Existence

AbstractLet M be a Riemannian manifold equipped with two complementary orthogonal distributions D and D ⊥. We introduce the conformal flow of the metric restricted to D with the speed proportional to the divergence of the mean curvature vector H, and study the question: When the metrics converge to one for which D enjoys a given geometric property, e.g., is harmonic, or totally geodesic? Our main observation is that this flow is equivalent to the heat flow of the 1-form dual to H, provided the initial 1-form is D ⊥-closed. Assuming that D ⊥ is integrable with compact and orientable leaves, we use known long-time existence results for the heat flow to show that our flow has a solution converging to a metric for which H = 0; actually, under some topological assumptions we can prescribe the mean curvature H.

scholarly journals Deforming a Convex Hypersurface by Anisotropic Curvature Flows

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
HongJie Ju ◽  
BoYa Li ◽  
YanNan Liu
Keyword(s):  
Principal Curvature ◽  
Support Function ◽  
Curvature Flow ◽  
Convex Hypersurface ◽  
Minkowski Problem ◽  
Smooth Solutions ◽  
Long Time Existence ◽  
Long Time ◽  
Anisotropic Curvature ◽  
Time Existence

AbstractIn this paper, we consider a fully nonlinear curvature flow of a convex hypersurface in the Euclidean 𝑛-space. This flow involves 𝑘-th elementary symmetric function for principal curvature radii and a function of support function. Under some appropriate assumptions, we prove the long-time existence and convergence of this flow. As an application, we give the existence of smooth solutions to the Orlicz–Christoffel–Minkowski problem.

scholarly journals On short time existence of Lagrangian mean curvature flow

2016 ◽  
Vol 367 (3-4) ◽  
pp. 1473-1515 ◽  
Author(s):  
Tom Begley ◽  
Kim Moore
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Short Time ◽  
Time Existence
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