# The normalized curve shortening flow and homothetic solutions

@article{Abresch1986TheNC, title={The normalized curve shortening flow and homothetic solutions}, author={Uwe Abresch and James Langer}, journal={Journal of Differential Geometry}, year={1986}, volume={23}, pages={175-196} }

The curve shortening problem, by now widely known, is to understand the evolution of regular closed curves γ: R/Z -> M moving according to the curvature normal vector: dy/dt = kN = -"the ZΛgradient of arc length". One motivation for this problem has been the view expressed in this connection by C. Croke, H. Gluck, W. Ziller, and others: it would be desirable to improve on some complicated and ad hoc constructions that have been used in the theory of closed geodesies to iteratively shorten…

## 302 Citations

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In this thesis we consider closed, embedded, smooth curves in the plane whose local total curvature does not lie below −π and study their behaviour under the area preserving curve shortening flow…

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κ ds L ≕ κ̄ is the average of the curvature. For simple closed curves ̄ κ = 2π L holds. Gage pointed out that this evolution equation arises as the “ L2-gradient flow” of the length functional under…

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We consider the curve shortening flow applied to a natural class of figureeight curves, those with dihedral symmetry and some monotonicity assumptions on the curvature and its derivatives. We prove…

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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…

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We show that the solutions to the curvature flow (CF) for curves on the 2-dimensional light cone are in correspondence with the solutions to the inverse curvature flow (ICF). We prove that the…

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is satisfied. Here H(p,ή is the mean curvature vector of the hypersurface Mt at F(/?, t). We saw in [7] that (1) is a quasilinear parabolic system with a smooth solution at least on some short time…

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The motion by curvature of networks is the generalization to finite union of curves of the curve shortening flow. This evolution has several peculiar features, mainly due to the presence of junctions…

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