Alternative Harnack Inequality for Curve Shortening Flow

• The paper presents a novel formulation that yields explicit time-dependent curvature estimates and rigidity via alternative Harnack inequalities.
• Methodologies incorporate algorithmic constructions, isoperimetric profile comparisons, and area-turning functionals applicable to both convex and nonconvex curves.
• The approach achieves sharp curvature control, convergence rates, and identifies delayed regularity phenomena, providing practical tools for singularity analysis.

The alternative Harnack inequality for curve shortening flow refers to a class of global and local differential inequalities controlling the evolution of geometric quantities along the curve shortening flow (CSF) in the plane, particularly in regimes extending beyond convex curves. This concept has seen substantial development, including algorithmic formulations for classical Harnack inequalities, isoperimetric profile methods for normalized flows, and new non-convex variants utilizing two-point functionals and global bounds. These methodologies yield explicit time-dependent curvature estimates, delayed regularity phenomena, and sharp control of geometric behavior for a broad class of initial curves.

1. Classical and Algorithmic Harnack Inequalities for CSF

Hamilton’s classical Harnack estimate for CSF establishes a pointwise differential inequality for the curvature $\kappa$ of a closed convex planar curve evolving by $\partial_t X = \kappa \nu$. The formulation is: $$\kappa_t + \frac{1}{2t}\kappa \geq \frac{\kappa_s^2}{\kappa}$$ where $s$ is the arclength parameter (Băileşteanu, 2015). The algorithmic approach involves constructing a functional (“Harnack quantity”) of the form $H = a\,u_{ss} + b\,u_s^2 + c\,e^{2u} + \phi(s, t)$ for $u := \log \kappa$, with suitable choices of $a, b, c, \phi$ ensuring positivity via the maximum principle.

Key features of this approach include:

• Generation of a one-parameter family of $\alpha$-Harnack inequalities: $$\kappa_t + \frac{\alpha\,\kappa}{t} \geq \frac{\kappa_s^2}{\kappa}, \quad \alpha \geq \frac12,\ t>0,\ \kappa>0$$
• The sharp constant $\alpha = 1/2$ realized by the shrinking circle soliton.
• Gradient and singularity estimates.
• Direct analysis of rigidity properties and curvature control.

2. Isoperimetric Profile Comparison and Global Alternative Inequality

An alternative “global” Harnack-type inequality emerges from comparison theorems for isoperimetric profiles under the normalized curve shortening flow (NCSF) (Andrews et al., 2011). Here, the evolution is rescaled so that curve-enclosed area remains constant, with dynamics: $$\frac{\partial X}{\partial t} = X + \frac1{|X'|}\Bigl(\frac{X'}{|X'|}\Bigr)' = X - \mathbf n$$ The isoperimetric profile $\Psi(\Omega, a)$ associates to each area $a$ the minimal length of inner boundaries enclosing that area. The profile satisfies asymptotics: $$\Psi(\Omega, a) = \sqrt{2\pi a} - \frac{4 \sup_{\partial\Omega} \kappa}{3\pi} a + O(a^{3/2}), \quad a \to 0$$ and comparison theorems provide bounds for the flow:

• If initially $\Psi(\Omega_0, a) \geq \Psi(\Theta_{t_0}, a)$, with $\Theta$ a model convex region, then for all $t \geq 0$

$\Psi(\Omega_t, a) \geq \Psi(\Theta_{t-t_0}, a)$

• Analogous bounds hold for the exterior profile for lower curvature estimates.

The central ODE governing the comparison function $f(a, t)$ is: $$\frac{\partial f}{\partial t} = -\frac{1}{f}\,\mathcal{F}(f\,f', f^3 f'') + f + f' (\pi - 2a) - f (f')^2$$ This ODE acts as a “differential Harnack-type” inequality for the profile, with pointwise curvature bounds extracted via profile asymptotics.

3. Non-Convex Alternative Harnack Inequality: Swept-Area and Turning Functionals

Recent developments have extended Harnack-type inequalities to arbitrary embedded planar curves with no convexity assumption, provided both ends are initially radial lines (Sobnack et al., 20 Jan 2026). This utilizes two-point global functionals:

• Swept area $V(v, w, t)$: $$V(v, w, t) := \frac{1}{2} \int_v^w * (X_u \wedge X) \, du$$
• Turning function $\Psi(v, w, t)$: $\Psi(v, w, t) := \psi(w, t) - \psi(v, t)$ with $\psi(u, t)$ the tangent angle.

Defining the Harnack quantity

$H(v, w, t) := V(v, w, t) - t\, \Psi(v, w, t)$

the main theorem establishes that, for suitable initial conditions,

$$V_- \leq H(v, w, t) \leq V_+, \quad \forall\ v<w,\ t\geq 0$$

where $V_\pm$ are initial area bounds. This provides global control of area and turning under CSF regardless of convexity.

4. Rigorous Proof Techniques and Boundary Regularity

The proof structures for these alternative Harnack inequalities involve:

• Evolution equations for $V$ and $\Psi$ showing that $H$ satisfies a two-point heat equation.
• Extension to parabolic boundaries via asymptotic analysis of the ends.
• Application of the strong maximum principle in the $(v, w)$ domain, utilizing boundary Dirichlet data and diagonal vanishing ($H(v, v, t) \equiv 0$) (Sobnack et al., 20 Jan 2026).
• Rigidity lemmas confirm the structure of optimizers (e.g., isoperimetric minimizers with connected inner boundary, constant curvature arcs).
• Second-variation and time-variation inequalities yield sharp curvature bounds directly, bypassing blow-up analysis and compactness arguments.

5. Sharp Curvature Bounds and Delayed Regularity Phenomena

Explicit upper and lower curvature bounds follow from the isoperimetric-profile method (Andrews et al., 2011): $$-\;C_1\,e^{-t} \leq \kappa(x, t) \leq 1 + C_2\,e^{-2t}, \quad x \in \partial\Omega_t,\ t \geq 0$$ for suitable constants $C_1, C_2$. For “wild” initial curves with radial ends, the alternative Harnack inequality quantifies the delay in regularization: $t > \frac{2(V_+ - V_-)}{\alpha}$ implies the flow straightens and becomes globally graphical, with uniform $C^k$ bounds attained after this explicit time (Sobnack et al., 20 Jan 2026).

For polar graphical flows, the area–turning Harnack shows convergence to self-similar wedge expanders with asymptotic bounds: $$\big|\Psi(\theta_0, \theta_1, t) - \Psi^{\rm wedge}(\theta_0, \theta_1)\big| \leq \frac{V_0}{t}$$ implying the approach to the expander is linear in $t^{-1}$.

6. Connections, Comparative Analysis, and Applications

The alternative Harnack inequality generalizes the classical convex case. For convex flows, both the Hamilton and area–turning Harnack imply identical sharp curvature-support bounds: $$k(p, t) \leq \frac{1}{2 t} D(p, t),\quad D(p, t) = \langle X(p, t), -\nu(p, t)\rangle$$ with sharp equality on the wedge expanders (Sobnack et al., 20 Jan 2026). The alternative Harnack methodology has connections to delayed regularization in Ricci flow and obstructions in Lagrangian mean curvature flow.

Applications include:

• Control of curvature growth, singularity analysis, and soliton rigidity.
• Quantification of parabolic regularity delay for non-convex flows.
• Explicit convergence rates for normalized flows.
• Frameworks for non-convex graphical flows with prescribed asymptotics.

The alternative Harnack inequalities thus provide unified, sharp, and global control over the evolution of curves under CSF, extending the analytic and geometric reach of Harnack-type analysis to arbitrary embedded initial data, and delivering explicit regularization times and geometry-driven bounds.