Curve shortening makes convex curves circular

@article{Gage1984CurveSM,
  title={Curve shortening makes convex curves circular},
  author={Michael E. Gage},
  journal={Inventiones mathematicae},
  year={1984},
  volume={76},
  pages={357-364}
}
  • M. Gage
  • Published 1 June 1984
  • Medicine
  • Inventiones mathematicae
View on Springer
269 Citations
Highly Influential Citations
25
Background Citations
104
Methods Citations
24
Results Citations
7

269 Citations

Computing critical point evolution under planar curvature flows

We present a numerical method for computing the evolution of a planar, star-shaped curve under a broad class of curvature-driven geometric flows, which we refer to as the Andrews-Bloore flows. This…

Whitney-Graustein Homotopy of Locally Convex Curves via a Curvature Flow

Let $X_0, \widetilde{X}$ be two smooth, closed and locally convex curves in the plane with same winding number. A curvature flow with a nonlocal term is constructed to evolve $X_0$ into…

Gage’s original normalized CSF can also yield the Grayson theorem

Mimicking Andrews-Bryan’s argument, it is proved in this note that Gage’s original normalized curve shortening flow can also yield the Grayson theorem.

A note on Grayson's theorem

In this note we show a variational proof of Matthew Grayson’s convexification theorem for simple closed curves moving by curvature in the plane. CONTENTS

Noncollapsing in mean-convex mean curvature flow

We provide a direct proof of a noncollapsing estimate for compact hypersurfaces with positive mean curvature moving under the mean curvature flow: Precisely, if every point on the initial…

Non-collapsing in mean-convex mean curvature flow

We provide a direct proof of a non-collapsing estimate for compact hypersurfaces with positive mean curvature moving under the mean curvature flow: Precisely, if every point on the initial…

The blow up analysis of the general curve shortening flow

It is shown that the curvature function satisfies a nonlinear evolution equation under the general curve shortening flow and a detailed asymptotic behavior of the closed curves is presented when they…

Evolution of an extremum by curvature motion

POLYGON SHORTENING MAKES (MOST) QUADRILATERALS CIRCULAR

We show that an analog of the Gage-Grayson-Hamilton Theorem for curves moving according to their mean curvature holds for the motion of quadrilaterals according to their Menger curva- ture.

Medial Axes and Mean Curvature Motion I: Regular Points

A set of conditions on the local validity of a medial axis transform and a differential equation for the change of smooth parts of the medial axis when its generating curve evolves under MCM are presented.
...

References

SHOWING 1-3 OF 3 REFERENCES

BONNESEN-STYLE ISOPERIMETRIC INEQUALITIES

Because of Property 1, any Bonnesen inequality implies the isoperimetric inequality (1). From Property 2, it follows that equality can hold in (1) only when C is a circle. The effect of Property 3 is…

Convex Sets and Their Applications

Fundamentals. Hyperplanes. Helly-Type Theorems. Kirchberger-Type Theorems. Special Topics in E2. Families of Convex Sets. Characterizations of Convex Sets. Polytopes. Duality. Optimization. Convex…

An isoperimetric inequality with applications to curve shortening