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The curve-shortening flow is an example of a geometric flow, and is the one-dimensional case of the mean curvature flow. Other names for the same process include the Euclidean shortening flow, geometric heat flow, [1] and arc length evolution. As the points of any smooth simple closed curve move in this way, the curve remains simple and smooth ...
In the mathematical fields of differential geometry and geometric analysis, the Gauss curvature flow is a geometric flow for oriented hypersurfaces of Riemannian manifolds. In the case of curves in a two-dimensional manifold, it is identical with the curve shortening flow.
Figure 1, is a generalized representation of the geometry assumed by a detachment fault. Figure 1. The general geometry of a detachment fold illustrating the shortening above a layer parallel decollement and the resulting geometry of a detachment fold in a compressional environment.
For surfaces of dimension two or more this is a theorem of Gerhard Huisken; [5] for the one-dimensional curve-shortening flow it is the Gage–Hamilton–Grayson theorem. However, there exist embedded surfaces of two or more dimensions other than the sphere that stay self-similar as they contract to a point under the mean-curvature flow ...
The book consists of four chapters, roughly divided into four sections: [1] Chapters 1 through 4 concern the heat equation and the curve-shortening flow defined from it, in which a curve moves in the Euclidean plane, perpendicularly to itself, at a speed proportional to its curvature. [1]
In the mathematical field of differential geometry, a geometric flow, also called a geometric evolution equation, is a type of partial differential equation for a geometric object such as a Riemannian metric or an embedding. It is not a term with a formal meaning, but is typically understood to refer to parabolic partial differential equations.
The curve-shortening flow decreases the isoperimetric ratio of any smooth convex curve so that, in the limit as the curve shrinks to a point, the ratio becomes 4 π. [ 2 ] For higher-dimensional bodies of dimension d , the isoperimetric ratio can similarly be defined as B d / V d − 1 where B is the surface area of the body (the measure of its ...
The Angenent torus can be used to prove the existence of certain other kinds of singularities of the mean curvature flow. For instance, if a dumbbell shaped surface, consisting of a thin cylindrical "neck" connecting two large volumes, can have its neck surrounded by a disjoint Angenent torus, then the two surfaces of revolution will remain disjoint under the mean curvature flow until one of ...
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