Isoperimetric inequalities for special classes of curves
The curve shortening problem is described by the partial differential equation \[ X_t=kN \tag{1} \] for a closed initial curve \(C(0)\) of class \(C^2\), where \(X\) is a parametrization of the evolving family of curves \(C(t)\), \(N\) is the inward unit normal vector, \(k\) is the curvature, and the subscript \(t\) denotes the partial derivative at time \(t\). Solutions of \((1)\) which do not change their shape during the evolution, are called homothetic. In the paper under review, the classical Banchoff-Pohl inequality, an isoperimetric inequality for nonsimple closed curves in the Euclidean plane, involving the square of the winding number, is sharpened for homothetic or Abresch-Langer solutions of curve shortening. For a larger class of curves and for rotationally symmetric curves, further isoperimetric inequalities containing the rotation number and the winding number are presented.
scientific article; zbMATH DE number 5880266
scientific article