A cable system defines an index agreeing with Brouwer degree on complementary regions, providing a sharp lower bound on swept volume for null homotopies of immersed spheres in R^3 that is attained under sense-preserving monotonic conditions.
On minimum area homotopies of normal curves in the plane
2 Pith papers cite this work. Polarity classification is still indexing.
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abstract
In this paper, we study the problem of computing a homotopy from a planar curve $C$ to a point that minimizes the area swept. The existence of such a minimum homotopy is a direct result of the solution of Plateau's problem. Chambers and Wang studied the special case that $C$ is the concatenation of two simple curves, and they gave a polynomial-time algorithm for computing a minimum homotopy in this setting. We study the general case of a normal curve $C$ in the plane, and provide structural properties of minimum homotopies that lead to an algorithm. In particular, we prove that for any normal curve there exists a minimum homotopy that consists entirely of contractions of self-overlapping sub-curves (i.e., consists of contracting a collection of boundaries of immersed disks).
fields
math.GT 2years
2026 2verdicts
UNVERDICTED 2representative citing papers
Existence of area-minimizing null homotopies for C^1 and Lipschitz immersed planar curves is shown by lifting to embedded curves, applying Douglas minimizers, and proving convergence back to the plane.
citing papers explorer
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Minimal Homotopies in Three Dimensions: A Cable System Approach
A cable system defines an index agreeing with Brouwer degree on complementary regions, providing a sharp lower bound on swept volume for null homotopies of immersed spheres in R^3 that is attained under sense-preserving monotonic conditions.
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Existence of Minimal Homotopies for Immersed Planar Curves
Existence of area-minimizing null homotopies for C^1 and Lipschitz immersed planar curves is shown by lifting to embedded curves, applying Douglas minimizers, and proving convergence back to the plane.