The Curve Shortening Flow

@inproceedings{Epstein1987TheCS,
  title={The Curve Shortening Flow},
  author={Charles L. Epstein and Michael E. Gage},
  year={1987}
}
This is an expository paper describing the recent progress in the study of the curve shortening equation $${X_{{t\,}}} = \,kN $$ (0.1) Here X is an immersed curve in ℝ2, k the geodesic curvature and N the unit normal vector. We review the work of Gage on isoperimetric inequalities, the work of Gage and Hamilton on the associated heat equation and the work of Epstein and Weinstein on the stable manifold theorem for immersed curves. Finally we include a new proof of the Bonnesen… 
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124 Citations
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124 Citations

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Figure Captions
1. Geometry of the geodesic curvature vector. 2. Geometry of the constant velocity projection. 3. A simple planar example. The nal geodesic curve is as straight line as expected. (a) Top view (the ^…
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References

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TLDR
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BONNESEN-STYLE ISOPERIMETRIC INEQUALITIES
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The heat equation shrinking convex plane curves
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