scholarly journals The Curve Shortening Flow in the Metric-Affine Plane

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 701
Author(s):  
Vladimir Rovenski
Keyword(s):  
Classical Result ◽  
Affine Plane ◽  
Euclidean Plane ◽  
Geometric Condition ◽  
Convex Curve ◽  
Initial Curve ◽  
Curve Shortening Flow ◽  
Convex Curves ◽  
Curve Shortening ◽  
First Time

We investigated, for the first time, the curve shortening flow in the metric-affine plane and prove that under simple geometric condition (when the curvature of initial curve dominates the torsion term) it shrinks a closed convex curve to a “round point” in finite time. This generalizes the classical result by M. Gage and R.S. Hamilton about convex curves in a Euclidean plane.

scholarly journals The area preserving curve shortening flow with Neumann free boundary conditions

2015 ◽  
Vol 1 (1) ◽  
Author(s):  
Elena Mäder-Baumdicker
Keyword(s):  
Free Boundary ◽  
Boundary Conditions ◽  
Convex Domain ◽  
Euclidean Plane ◽  
Initial Curve ◽  
Curve Shortening Flow ◽  
Free Boundary Conditions ◽  
Curve Shortening ◽  
The Given ◽  
Area Preserving

AbstractWe study the area preserving curve shortening flow with Neumann free boundary conditions outside of a convex domain in the Euclidean plane. Under certain conditions on the initial curve the flow does not develop any singularity, and it subconverges smoothly to an arc of a circle sitting outside of the given fixed domain and enclosing the same area as the initial curve.

A non-local expanding flow of convex closed curves in the plane

2020 ◽  
pp. 2050115
Author(s):  
Ke Shi
Keyword(s):  
Numerical Experiment ◽  
Additional Assumption ◽  
Euclidean Plane ◽  
Blow Up ◽  
Time Interval ◽  
Initial Curve ◽  
Closed Curves ◽  
Convex Curves ◽  
Non Local

This paper presents a new non-local expanding flow for convex closed curves in the Euclidean plane which increases both the perimeter of the evolving curves and the enclosed area. But the flow expands the evolving curves to a finite circle smoothly if they do not develop singularity during the evolving process. In addition, it is shown that an additional assumption about the initial curve will ensure that the flow exists on the time interval [Formula: see text]. Meanwhile, a numerical experiment reveals that this flow may blow up for some initial convex curves.

scholarly journals TRANSLATING SOLITONS FOR LAGRANGIAN MEAN CURVATURE FLOW IN COMPLEX EUCLIDEAN PLANE

2012 ◽  
Vol 23 (10) ◽  
pp. 1250101 ◽  
Author(s):  
ILDEFONSO CASTRO ◽  
ANA M. LERMA
Keyword(s):  
Mean Curvature ◽  
Euclidean Plane ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Curve Shortening Flow ◽  
Self Similar ◽  
Curve Shortening ◽  
Similar Solutions ◽  
Hamiltonian Stationary ◽  
Translating Solitons

Using certain solutions of the curve shortening flow, including self-shrinking and self-expanding curves or spirals, we construct and characterize many new examples of translating solitons for mean curvature flow in complex Euclidean plane. They generalize the Joyce, Lee and Tsui ones [Self-similar solutions and translating solitons for Lagrangian mean curvature flow, J. Differential Geom.84 (2010) 127–161] in dimension two. The simplest (non-trivial) example in our family is characterized as the only (non-totally geodesic) Hamiltonian stationary Lagrangian translating soliton for mean curvature flow in complex Euclidean plane.

Area properties associated with a convex plane curve

2017 ◽  
Vol 24 (3) ◽  
pp. 429-437
Author(s):  
Dong-Soo Kim ◽  
Young Ho Kim ◽  
Hyeong-Kwan Ju ◽  
Kyu-Chul Shim
Keyword(s):  
Characteristic Property ◽  
Plane Curve ◽  
Characterization Theorem ◽  
Convex Curve ◽  
Convex Plane ◽  
Convex Curves ◽  
Necessary And Sufficient ◽  
Locally Convex

AbstractArchimedes knew that for a point P on a parabola X and a chord AB of X parallel to the tangent of X at P, the area of the region bounded by the parabola X and chord AB is four thirds of the area of the triangle {\bigtriangleup ABP}. Recently, the first two authors have proved that this fact is the characteristic property of parabolas.In this paper, we study strictly locally convex curves in the plane {{\mathbb{R}}^{2}}. As a result, generalizing the above mentioned characterization theorem for parabolas, we present two conditions, which are necessary and sufficient, for a strictly locally convex curve in the plane to be an open arc of a parabola.

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