Homogeneous Sobolev gradient flow of the length functional

1. Problem Statement

The paper investigates the gradient flow of the length functional $L(\gamma)$ on the space of planar immersed closed curves, denoted as $\text{Imm}^1(S, \mathbb{R}^2)$. Unlike the classical Curve Shortening Flow (which uses the $L^2$ metric and leads to finite-time singularities for general curves) or the inhomogeneous Sobolev flow studied previously by the authors, this work focuses on homogeneous Sobolev $H^1$-type Riemannian metrics.

The specific metric considered is:
$$\langle v, w \rangle_{H^1_{\lambda,a}(\gamma)} := \frac{1}{L(\gamma)^a} \langle v, w \rangle_{L^2(\gamma)} + \frac{\lambda^2}{L(\gamma)^{a-2}} \left\langle \frac{v'}{|\gamma'|}, \frac{w'}{|\gamma'|} \right\rangle_{L^2(\gamma)}$$
where:

• $\lambda > 0$ is a scaling parameter.
• $a \in \mathbb{R}$ is a homogeneity parameter.
• The powers of $L(\gamma)$ ensure the metric is $(3-a)$-homogeneous under dilations of the plane.

The goal is to analyze the gradient flow equation $\partial_t X = -\nabla L(X)$ defined by this metric, specifically addressing existence, regularity, long-time behavior, and geometric preservation properties (immersion and convexity).

2. Methodology

A. Explicit Gradient Derivation

The authors derive an explicit formula for the gradient $\nabla L_{\lambda,a}(\gamma)$. By utilizing the periodic Green's function $G_\lambda$ associated with the operator $(\lambda^2 \partial_x^2 - 1)$, the gradient is expressed as a nonlocal convolution:
$$\nabla L_{\lambda,a}(\gamma) = \frac{L(\gamma)^{a-2}}{\lambda^2} \left( \gamma + \gamma *_\gamma G_\lambda \right)$$
where $*_\gamma$ denotes a convolution defined with respect to the arc-length measure of the curve. This formulation reveals the flow as a reparametrisation-invariant nonlocal ODE.

B. Low-Regularity Framework

The analysis is conducted in the optimal low-regularity space $W^{1,1}(S, \mathbb{R}^2)$. This is the largest space where the length functional is well-defined and the gradient operator satisfies the continuity properties required for the Picard–Lindelöf theorem.

• Local Well-Posedness: The authors prove that the gradient vector field is locally Lipschitz continuous on $W^{1,1}$ away from constant maps (where length is zero). This allows for the application of the Picard–Lindelöf theorem to guarantee local existence and uniqueness.

C. Time Reparametrisation

A crucial methodological step is the reduction of the general case ($a \in \mathbb{R}$) to the model case $a=2$.

• The flow for general $a$ is shown to be equivalent to the flow for $a=2$ via a time reparametrisation $\tau(t)$ governed by the length $L(t)$.
• Specifically, if $X$ solves the flow for $a=2$, a reparametrised version solves the flow for any $a$. This allows the authors to focus their qualitative analysis (decay, convergence, preservation of geometry) on the $a=2$ case and extend results to all $a$.

D. Geometric Analysis

For higher regularity initial data ($C^1$ and $C^2$), the authors employ:

• Maximum Principle arguments (via Danskin's theorem) to control the $L^\infty$ norm and derivatives.
• Curvature evolution equations to analyze the preservation of convexity.
• Gronwall's inequality to prevent blow-up of norms in finite time, ensuring global existence.

3. Key Contributions and Results

I. Local and Global Well-Posedness

• Theorem 1 (i): For any initial curve $X_0 \in W^{1,1}(S, \mathbb{R}^2)$ with positive length, there exists a unique local solution $X \in C^1([0, T_0], W^{1,1})$.
• Theorem 1 (ii) & (iv): By reducing to the $a=2$ case, the authors prove global existence for all $t \ge 0$. The solution converges uniformly and in the $W^{1,1}$ norm to a constant map (a point).

II. Decay of Length

• Theorem 1 (iii): The length of the curve decays exponentially for the $a=2$ case:
  $$L(X(t)) \le L(X_0) e^{-4t/(1+8\lambda^2)}$$
• Extinction Time:
  • If $a < 2$: The curve collapses to a point in finite time.
  • If $a \ge 2$: The curve collapses to a point in infinite time.

III. Preservation of Geometric Properties

A significant novelty of this homogeneous flow compared to previous inhomogeneous models is the preservation of convexity.

• Preservation of Immersion (Theorem 2, v): If the initial curve is a $C^1$ immersion ($|\gamma'| \neq 0$), it remains an immersion for all time.
• Preservation of Convexity (Theorem 2, vi): If the initial curve is a $C^2$ immersion bounding a strictly convex domain, the flow preserves this convexity for all time. This is proven by showing that the curvature $k$ remains strictly positive, satisfying $k(u,t) \ge k(u,0)e^{-t/\lambda^2}$.

IV. Numerical Observations

The authors provide numerical simulations (using a discretized convolution scheme) demonstrating:

• Exponential decay of length.
• A "rounding" effect where non-circular curves evolve toward a circular shape.
• Asymptotic behavior suggests the solution approaches a round circle (up to scaling) regardless of $\lambda$, consistent with the theoretical collapse to a point.

4. Significance

1. Resolution of Metric Pathologies: The paper addresses the issue that $L^2$ metrics on shape spaces induce a Riemannian distance of zero (as shown by Michor and Mumford). By using homogeneous Sobolev metrics, the authors establish a flow with non-degenerate geometric properties.
2. Convexity Preservation: While many geometric flows (like Curve Shortening Flow) preserve convexity, the authors demonstrate that this specific nonlocal Sobolev gradient flow also preserves convexity. This is a non-trivial result given the nonlocal nature of the operator (convolution with Green's function).
3. Optimal Regularity: The analysis in $W^{1,1}$ is significant because it pushes the regularity requirements to the theoretical minimum, making the results applicable to a broader class of initial data than standard $C^k$ flows.
4. Scale Invariance: The homogeneous nature of the metric (specifically the $a=2$ case) ensures the flow is equivariant under Euclidean scaling, making it a natural candidate for shape analysis and computer vision applications where scale invariance is desired.

In summary, the paper establishes a rigorous mathematical framework for a scale-invariant, nonlocal gradient flow of curve length, proving its well-posedness in low regularity spaces and demonstrating its ability to smooth curves while preserving essential geometric features like immersion and convexity.