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 The Minimum Area Of Null Homotopies Of Curves Traced Twice
2 Pith papers cite this work. Polarity classification is still indexing.
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Pith papers citing it
abstract
We provide an efficient algorithm to compute the minimum area of a homotopy between two closed plane curves, given that they divide the plane into finite number of regions. For any positive real number $\varepsilon>0$, we construct a closed plane curve $\gamma$ such that the minimum area of a null homotopy of $2\cdot\gamma$ is less than $\varepsilon$ times that of $\gamma$. We also establish a lower bound on how complex a desired closed curve has to be.
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.