Absolute Variation Curvature: Theory & Applications

• Absolute Variation Curvature is a measure quantifying the total geometric bending of curves and submanifolds, including singular and non-smooth features.
• It extends classical curvature concepts through BV analysis and measure theory, accommodating frontals and distributional curvature measures.
• Its applications span magnetic geometry, variational imaging, and geometric evolution equations, preserving key curvature bounds and convexity properties.

Absolute variation curvature quantifies the total geometric bending of curves or submanifolds, robustly accommodating singularities and non-smooth features. It finds rigorous expression in modern geometric analysis, the theory of weak (BV) curves, and applications such as magnetic geometry and variational imaging. Central to recent advances is the extension of classical total curvature—typically defined for smooth immersions—to more general objects, including frontals with singularities and distributional (BV) curves. The resulting framework preserves critical results such as curvature bounds and convexity characterizations while enabling new applications in settings with nontrivial singular structure.

1. Definitions and Foundational Notions

Absolute variation curvature arises from the total variation of generalized tangent or normal fields associated to curves or submanifolds. For a Lipschitz (rectifiable) curve $\gamma:[0,L]\to\mathbb{R}^n$ parameterized by arc-length, the unit tangent $T:[0,L]\to S^{n-1}$ belongs to the space of functions of bounded variation (BV) if and only if its distributional derivative $DT$ is a finite Radon measure. The curvature measure is defined by $\kappa\,ds := |DT|$, so the total absolute curvature is

$$\text{TotCurv}(\gamma) := |DT|([0,L]) = \int_0^L \kappa(s)\,ds$$

When $\gamma$ is $C^2$ this reduces to $\int_0^L |T'(s)|ds$. For planar curves parameterized by tangent angle $\theta(s)$, $\text{TotCurv}(\gamma) = \int_0^L |\theta'(s)|ds$; for piecewise linear curves, it equals the sum of turning angles at vertices, recovering Milnor's notion (Mucci et al., 14 Jun 2025).

For higher-dimensional admissible frontals $f:M^n\to\mathbb{R}^{n+r}$, the total absolute curvature is

$$\tau(M, f) = \frac{1}{\text{Vol}\,S^{n+r-1}} \left( \int_B |G|\,d\mu_B + \int_{\bar{B}} |\bar{G}|\,d\mu_{\bar{B}} \right)$$

where $G(p,\xi)$ is the determinant of the shape operator (Lipschitz–Killing curvature) at $(p,\xi)$ in the unit normal bundle $B$, and the second term covers the singular set (Yamauchi, 26 Dec 2024).

In the context of magnetic field lines, the absolute variation curvature $k_m$ of the field line unit vector is defined by

$$k_m = \frac{1}{2}\,\mathbf{b}\cdot\nabla\ln B = \mathbf{b}\cdot\nabla\ln \sqrt{B}$$

and quantifies the rate of variation of $|B|$ (magnetic field strength) along its lines (Skovoroda et al., 2010).

2. Curves with Singularities: Frontals and Curvature Measures

Curves with densely regular points but possible isolated singularities (where the velocity vanishes) are modeled as frontals. Such $\gamma:I\to\mathbb{R}^n$ admit a unit tangent field $e:I\to S^{n-1}$ that is everywhere linearly dependent with $\gamma'$. For regular points, the classical curvature function remains meaningful, but at singularities the vector-valued curvature may blow up. The key insight is that the 1-form $k\,ds := \|e'(t)\|dt$ extends continuously across singularities, yielding a well-defined curvature measure.

A non-co-orientable closed frontal ($e(t+2\pi) = -e(t)$) in $\mathbb{R}^n$ satisfies a generalized Fenchel theorem:

$K(\gamma) := \int_{S^1} k\,ds \geq \pi$

with equality if and only if the curve lies in a plane, is locally $L$-convex, and has rotation index $\pm 1/2$. For such a planar frontal $\gamma$, the angle function $\theta(t)$ determines the signed curvature; $K(\gamma) = \int_0^{2\pi} |\theta'(t)|dt$, and rotation index $$\mathrm{ind}_\gamma = [\theta(2\pi) - \theta(0)]/(2\pi)$$ is half-integral in the non-co-orientable case (Honda et al., 1 Mar 2024).

Singularities classified as ordinary cusps (locally like $t\mapsto (t^2, t^3)$) obey further constraints: in the equality case $K(\gamma)=\pi$, the number $N$ of cusps is an odd integer $\geq 3$; $N=3$ if and only if the curve is a simple closed curve.

3. Extension to Higher Dimensions: Admissible Frontals and Submanifolds

The total absolute curvature for an admissible compact frontal submanifold $f:M^n\to\mathbb{R}^{n+r}$ adapts the classical (Chern–Lashof) theory to singular settings. Co-orientability requires a globally defined lifted Gauss map into the oriented Grassmannian. The regular points contribute Lipschitz–Killing curvature via the determinant of the shape operator, while singularities of the first kind are incorporated by integrating the associated curvature over the singular locus. The sum of these measures yields $\tau(M, f)$.

A key result is the generalized Chern–Lashof inequality:

$\tau(M, f) \geq \sum_{i=0}^n b_i(M)$

where $b_i(M)$ are Betti numbers. If $\tau=2$ and all singularities are of the first kind, the frontal is the boundary of a convex domain in an affine $n$-plane (Yamauchi, 26 Dec 2024).

This framework allows precise comparison to critical point theory (Morse functions on $M$), ensuring the integral-geometric content of total curvature is preserved even in the presence of singularities.

4. Absolute Variation Curvature in Physical and Computational Contexts

In magnetized plasma physics, absolute variation curvature $k_m$ of the magnetic field lines quantifies the precise rate at which the field strength $|B|$ changes along the field direction. The fundamental relation

$\mathbf{b} \cdot \nabla B = 2 k_m B$

shows that $k_m$ measures the normalized gradient of $\log B$ along $\mathbf{b}$, i.e., how absolute value varies along trajectories. Surfaces of constant mean curvature optimize volume-to-surface-area ratio in confinement geometries, maximizing plasma confinement time due to this curvature property (Skovoroda et al., 2010).

In experimental optics, absolute wavefront curvature (radius of curvature of a Gaussian beam) is essential for differential wavefront sensing in heterodyne interferometers. A robust two-lever-arm measurement is provided to directly solve for the absolute curvature radius $R$ from measured phase-difference slopes, with analytic sensitivity estimates. The method is stable to longitudinal noise and enables $\sim 0.08$ m precision under typical parameters (Hechenblaikner, 2010).

In variational image analysis, classical absolute curvature energies are lifted to the roto-translation space, providing convex relaxation models of “total roto-translational variation” that tightly recover the geometric |$\kappa$|–energy for characteristic functions of smooth sets. These techniques are effective in image denoising and shape regularization (Chambolle et al., 2017).

5. Weak and Distributional Frameworks

The measure-theoretic approach generalizes the Frenet–Serret theory to curves whose tangent and normal are functions of bounded variation. A generalized tangent field $T \in BV([0, L];\mathbb{R}^3)$ has a curvature measure $\kappa ds = |DT|$. The distributional Frenet system

$$D_s T = \kappa N, \qquad D_s N = -\kappa T + \tau B, \qquad D_s B = -\tau N$$

holds in the sense of Radon measures, with torsion $\tau ds$ defined by the off-diagonal part of the differential of the moving frame. The total curvature and total absolute torsion are the respective total variations of these measures (Mucci et al., 14 Jun 2025).

For polygonal curves, the measure-theoretic total curvature equals the sum of exterior angles; for continuous curves, this recovers the classical interpretation. The existence and uniqueness theorem guarantees that any (BV) prescription of curvature and torsion data yields (up to rigid motion) a unique arc-length parameterized $C^1$ curve with finite total curvature and torsion.

6. Dynamical and Analytical Consequences

In geometric evolution equations, such as curve shortening flow (CSF), total absolute curvature behaves monotonically. For each time-evolved approximant, the L¹-norm of curvature is nonincreasing, both at the discrete and continuum levels. This monotonicity is crucial for global construction of solutions insensitive to finite singularities, as in the semi-discrete CSF scheme (Guidotti, 2023). Notably, at singular times where the curve develops cusps or sheds loops, the total absolute curvature jumps downward by $\pi$ per cusp, reflecting the dissipation and topological transitions encoded by the geometric norm.

The existence of such robust, lower-semicontinuous curvature functionals on the space of immersed curves (or their equivalence classes under reparameterization) is essential for compactness and regularity properties in variational and flow settings.

7. Applications and Outlook

Absolute variation curvature underpins a spectrum of modern advances:

• In geometric knot theory, it allows precise classification of curves and knots up to critical curvature bounds, especially in the Milnor total curvature paradigm.
• In singular geometry, it enables generalizations of the Fenchel and Chern–Lashof theorems encompassing frontals and submanifolds with isolated but generic singularities, determining sharp lower curvature bounds, equality characterizations, and cusp-count restrictions (Honda et al., 1 Mar 2024, Yamauchi, 26 Dec 2024).
• In magnetic geometry, it dictates the confinement properties of plasma devices and the design of surfaces with optimal mean/absolute curvature (Skovoroda et al., 2010).
• In optimization and imaging, it provides the analytic backbone for shape-driven variational priors and inpainting schemes that capture global continuity and curvature regularization beyond total variation (Chambolle et al., 2017).

Further directions include analogues in Lorentzian, sub-Riemannian, or CAT$(\kappa)$ geometries, and the application to curves and submanifolds in higher codimension or open cases with prescribed singular patterns. The absolute variation curvature thus unifies measure-theoretic, geometric, and computational perspectives in the quantification of bending, regularity, and topological constraints.