[Paper Review] Interpolation inequalities between the deviation of curvature and the isoperimetric ratio with applications to geometric flows

Takeyuki Nagasawa, Kohei Nakamura|arXiv (Cornell University)|Nov 26, 2018

Geometric Analysis and Curvature Flows5 citations

TL;DR

This paper establishes new interpolation inequalities linking the deviation of curvature and the isoperimetric ratio for closed plane curves, enabling refined analysis of geometric flows. It proves exponential convergence to a round circle for area-preserving curvature flow without convexity assumptions, using scale-invariant quantities and Gagliardo-Nirenberg-type estimates to control curvature deviation and isoperimetric ratio decay.

ABSTRACT

Several inequalities for the isoperimetric ratio for plane curves are derived. In particular, we obtain interpolation inequalities between the deviation of curvature and the isoperimetric ratio. As applications, we study the large-time behavior of some geometric flows of closed plane curves without a convexity assumption.

Motivation & Objective

• To derive interpolation inequalities connecting curvature deviation and isoperimetric ratio for closed plane curves.
• To analyze the large-time behavior of geometric flows without requiring initial convexity.
• To establish exponential decay estimates for the isoperimetric ratio and curvature deviation using scale-invariant quantities.
• To extend convergence results for area-preserving curvature flow to non-convex initial curves.
• To provide quantitative decay rates for geometric flows based on curvature and isoperimetric ratio.

Proposed method

• Define scale-invariant quantities $ I_{ ext{--}1} = 1 - \frac{4\pi A}{L^2} $ (isoperimetric ratio deviation) and $ I_\ell = L^{2\ell+1} \int_0^L |\tilde{\kappa}^{(\ell)}|^2 ds $ (curvature deviation norms).
• Establish improved inequality $ 0 \leq I_{-1} \leq \frac{I_0}{8\pi^2} $, refining the crude bound $ I_{-1} \leq I_0^{1/2} $.
• Use the Gagliardo-Nirenberg inequality to interpolate $ I_\ell $ between $ I_{-1} $ and $ I_m $ for $ 0 \leq \ell \leq m $, yielding $ I_\ell \leq C\left( I_{-1}^{\frac{m-\ell}{2}} I_m + I_{-1}^{\frac{m-\ell}{m+1}} I_m^{\frac{\ell+1}{m+1}} \right) $.
• Apply the inequalities to area-preserving curvature flow $ \partial_t \mathbf{f} = \tilde{\kappa} \boldsymbol{\nu} $, deriving exponential decay of $ I_{-1} $ and $ I_0 $.
• Use complex Fourier expansion of the curve to relate curvature derivatives to $ I_\ell $, enabling $ L^2 $-based estimates.
• Combine energy estimates and isoperimetric inequalities to prove $ \frac{d}{dt}(L^2 - 4\pi A) \leq -\frac{16\pi^2}{L(0)^2}(L^2 - 4\pi A) $, implying exponential decay.

Experimental results

Research questions

• RQ1 Can interpolation inequalities be established between the deviation of curvature and the isoperimetric ratio for closed plane curves?
• RQ2 How can these inequalities be used to analyze the long-time behavior of geometric flows without convexity assumptions?
• RQ3 What is the precise exponential decay rate of the isoperimetric ratio for area-preserving curvature flow?
• RQ4 Can the curvature deviation $ I_0 $ be controlled by the isoperimetric ratio $ I_{-1} $ and higher-order curvature norms?
• RQ5 Does the area-preserving curvature flow converge to a round circle even for non-convex initial curves?

Key findings

• The inequality $ 0 \leq I_{-1} \leq \frac{I_0}{8\pi^2} $ holds, improving the naive bound $ I_{-1} \leq I_0^{1/2} $.
• For $ 0 \leq \ell \leq m $, the interpolation inequality $ I_\ell \leq C\left( I_{-1}^{\frac{m-\ell}{2}} I_m + I_{-1}^{\frac{m-\ell}{m+1}} I_m^{\frac{\ell+1}{m+1}} \right) $ is established with a universal constant $ C $ independent of $ \tilde{\kappa} $ and $ L $.
• For the area-preserving curvature flow, the isoperimetric ratio decays exponentially: $ L^2 - 4\pi A \leq (L(0)^2 - 4\pi A(0)) \exp\left(-\frac{16\pi^2}{L(0)^2}t\right) $.
• The length converges to $ 2\sqrt{\pi A(0)} $ with rate $ \left|L(t) - 2\sqrt{\pi A(0)}\right| \leq \frac{L(0)^2 - 4\pi A(0)}{4\sqrt{\pi A(0)}} \exp\left(-\frac{16\pi^2}{L(0)^2}t\right) $.
• The center of mass and the curve itself converge to a circle with exponential rate, as shown via complex Fourier analysis and divergence theorem estimates.
• The results extend to other geometric flows, with all claims in Theorem 4.3 holding for global solutions of the area-preserving flow without convexity.