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Two nonlocal inverse curvature flows of convex closed plane curves

Zezhen Sun

TL;DR

The paper studies two new $1/κ^{n}$-type nonlocal curvature flows for convex closed planar curves, where the normal speed is $F(κ)-λ(t)$ with $F(κ)=κ^{-n}$. It derives the governing evolution equations, reformulates the problem in terms of $ρ$ and $ν=ρ^{n}$, and proves uniform convexity and long-time existence under a non-blow-up assumption. The main result shows that if the curvature remains bounded for all finite times, the flow exists for all time and the evolving curves converge in $C^{\infty}$ to a circle, with the isoperimetric deficit decaying exponentially. This provides a rigorous mechanism for smoothing convex curves to circular shapes via nonlocal inverse-curvature dynamics, extending the theory of nonlocal geometric flows with explicit convergence rates.

Abstract

In this paper we introduce two $1/κ^{n}$-type ($n\ge1$) curvature flows for closed convex planar curves. Along the flows the length of the curve is decreasing while the enclosed area is increasing. And finally, the evolving curves converge smoothly to a finite circle if they do not develop singularity during the evolution process.

Two nonlocal inverse curvature flows of convex closed plane curves

TL;DR

The paper studies two new -type nonlocal curvature flows for convex closed planar curves, where the normal speed is with . It derives the governing evolution equations, reformulates the problem in terms of and , and proves uniform convexity and long-time existence under a non-blow-up assumption. The main result shows that if the curvature remains bounded for all finite times, the flow exists for all time and the evolving curves converge in to a circle, with the isoperimetric deficit decaying exponentially. This provides a rigorous mechanism for smoothing convex curves to circular shapes via nonlocal inverse-curvature dynamics, extending the theory of nonlocal geometric flows with explicit convergence rates.

Abstract

In this paper we introduce two -type () curvature flows for closed convex planar curves. Along the flows the length of the curve is decreasing while the enclosed area is increasing. And finally, the evolving curves converge smoothly to a finite circle if they do not develop singularity during the evolution process.
Paper Structure (6 sections, 15 theorems, 73 equations)

This paper contains 6 sections, 15 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. The proof of Theorem \ref{['them']}
  3. Some basic evolution equations
  4. Uniform convexity of the evolving curve
  5. Long time existence
  6. Convergence of the flow under the long time existence assumption

Key Result

Theorem 1.1

Assume $n\ge1$ and $X_{0}(u)$ is a smooth convex closed curve.Consider the flow flow1(or flow2) and assume that the curvature $\kappa$ will not blow up to $\infty$ in any finite time during the evolution process. Then the flow exists for all time $t\in[0,\infty)$. The length $L(t)$ is decreasing whi

Theorems & Definitions (29)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 19 more