The curve shortening flow deforms a closed curve γ in the direction of its normal vectors. Precisely, one deforms γ by the evolution equation[special characters omitted] where κ is the curvature and n is the normal vector; the derivative of the tangent vector T with respect to arclength s is T'(s) = κn.
In the first part of the thesis, we will study the evolution of convex, closed curve γ in [special characters omitted]. We will study the theorem of Tso, which says that γ becomes analytic in short time, and converges to a point within finite time. We then explain the Gage theorem on the asymptotic behavior of γ: if the area enclosed by γ is rescaled to be a constant (e.g., π) then γ converges smoothly to a unit circle.
In local coordinates, the curve shortening flow (of a curve in an n-dimensional manifold M) can be written as[special characters omitted]here s is the arclength parameter , we write γ = (γ1, γ2, ..., γ n) in local coordinate system; and [special characters omitted] are the Christoffel symbols of the connection on the manifold M.
In the second part of this thesis, we use the same equation (0.1), but now take s to be a fixed parameter on γ; so s is not an arclength parameter of γ when t > 0. This modified equation is called the one-dimensional harmonic heat flow. On general manifolds of nonpositive curvature, under the harmonic heat flow, any loop converges to a minimal geodesic loop as t → ∞. As a special case, in Euclidean spaces, all loops, convex or not, will shrink to a point. We will present a proof of this fact.
|School:||California State University, Long Beach|
|School Location:||United States -- California|
|Source:||MAI 47/06M, Masters Abstracts International|
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