Linear Semi-Discrete Polyharmonic Flows of Closed Polygons

Abstract In 2007, Chow and Glickenstein considered a linear semi-discrete analogue of the second-order curve shortening flow for smooth closed curves. In this article, we consider linear semi-discrete analogues of the polyharmonic curve diffusion flows for curves in $${\mathbb {R}}^p,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msup> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> $$p\ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . Since our flows correspond to first-order systems of linear ordinary differential equations with constant coefficients, solutions can be written down explicitly. As an application of similar ideas, we consider a linear semi-discrete answer to Yau’s question of when one can flow one curve to another by a curvature flow. In this setting, we are able to flow any closed polygonal curve to any other with the same or differing number of vertices, in the sense of exponential convergence in infinite time to a translate of the target polygon.