Linear Semi-discrete Polyharmonic Flows of Closed Polygons

TL;DR

The paper develops linear, semi-discrete polyharmonic flows for polygons in $\mathbb{R}^p$, defined by $\frac{dX}{dt}=(-1)^{m+1}M^mX$ with $M$ circulant, enabling explicit solutions via Fourier diagonalisation. It proves that, for any initial polygon, the flow exists for all time and exponentially converges to a point, with the rescaled limit being an affine image of a planar regular polygon, and that higher order $m$ accelerates convergence. The authors extend the framework to higher codimensions, showing analogous self-similar and asymptotic behavior governed by planar eigenmodes mapped through linear transforms, and they formulate a semi-discrete Yau difference flow that drives any $X$ to a fixed target polygon $Y$ (even with different vertex counts) in finite time. Overall, the work provides explicit, spectral characterisations of polygonal flows, demonstrating that increasing the polyharmonic order yields faster stabilization and enabling discrete analogues of curvature-flow questions. The results connect discretised geometric flows with classical smooth polyharmonic flows and highlight the role of circulant-M-based spectral theory in governing long-time dynamics.

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 $\mathbb{R}^p, p\geq 2$. 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.

Figures (4)

• Figure 1: Evolution of a pentagon under the semi-discrete polyharmonic flow for different values of $m.$ Distinct time steps of the evolution are shown superimposed over the initial polygon. The same time step values are used for each case of $m$.
• Figure 2: Evolution of a hexagon under the semi-discrete polyharmonic flow for different values of $m.$ Distinct time steps of the evolution are shown superimposed over the initial polygon. The same time step values are used for each case of $m$.
• Figure 3: Different cases of pentagons flowing to other pentagons under the semi-discrete Yau difference flow. In each case, selected time steps of the evolution are shown superimposed over the initial and target polygons. The target polygon is given by the dashed line.
• Figure 4: Examples of flowing from an initial polygon to a target polygon with a different number of vertices by the semi-discrete linear Yau difference flow. Selected time steps of evolution are shown superimposed over the initial and target polygons. The target polygon is given by the dashed line.

Theorems & Definitions (40)

• Definition 2.1
• Remark 2.1
• Definition 2.2
• Remark 2.2
• Definition 3.1
• Lemma 3.1
• proof
• Definition 3.2
• Lemma 3.2
• proof