Many questions regarding curve shortening remain outstanding.
With its careful exposition and complete guide to the literature, The Curve Shortening Problem provides not only an outstanding starting point for graduate students and new investigations, but a superb reference that presents intriguing new results for those already active in the field. Share this Title.
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Content Online - PAMQ 9 () No. 3 Article 6
Differential Geom. Zentralblatt MATH identifier Abresch, U. The normalized curve shortening flow and homothetic solutions. More by U.
More by J. L and A are the length of and the area of the lamina it encloses. Revision received May 27, GAGE 3. Isoperimetric inequality. LEMMA 1.
Mathematics > Differential Geometry
L x 2 ds Inequality B now follows directly from inequality A. A convex, closed curve which is symmetric through the origin satisfies inequality A. The symmetry of the curve dictates that the breadth of the curve in the direction determined by a normal N s is twice the support function p s. The desired result comes from integrating this inequality with respect to s and using the fact that p ds 2A.
For an ; closed, convex C curve it is possible to choose an origin so that inequality A holds. If the curve is piecewise C then inequality B holds Y s such that the line from X s to Y s bisects the convex lamina bounded by 7. Choose the origin to be the midpoint of the line from X S1 to Y s1 and let the x axis lie along this chord.
The Curve Shortening Problem
To complete the proof we denote by "YI and ,2the portions of the boundary curve lying above and below the x axis. The enclosed area A is the same as for the original curve because the x axis bisects the original lamina. Inequality B follows from lemma 1. Jacobowitz has observed that the bone shaped nonconvex curve in Figure shows that inequality 13 does not hold for all simple closed curves.
When e is small enough inequality B fails and an attempt to shorten such a curve by the process discussed in the next section leads to an initial increase in the isoperimetric ratio. GAGE 4. Curve shortening. If a closed curve evolves with time according to the evolution equation X gN then the derivative of the isoperimetric ratio is given by L2 x2ds- rr- The subscript indicates differentiation with respect to t.
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Deforming a curve along the gradient of its length functional leads to the evolution equation above.