Curve Shortening Flow

• Curve Shortening Flow is the process where an immersed curve evolves in the normal direction proportional to its curvature, often resulting in convexity and finite-time extinction.
• It encompasses classical smooth flows alongside weak formulations such as level-set and Brakke flows, providing rigorous frameworks to handle singularities.
• Applications span geometric PDEs, topology, convex geometry, and numerical analysis, with techniques like Huisken’s entropy ensuring convergence and stability.

Curve shortening flow (CSF) denotes the geometric evolution of an immersed curve in a Riemannian manifold by its curvature vector. Formally, if $X : S^1 \times [0,T) \to \mathbb{R}^d$, the flow is governed by $\partial_t X = kN$, where $k$ is the scalar curvature and $N$ the (inward) unit normal. In higher codimension or on general surfaces, the flow is replaced by the normal Laplacian of $X$ with respect to arclength. CSF has been central to geometric analysis, with applications to geometric PDEs, topology, convex geometry, and numerical analysis. Its study encompasses both classical smooth flows and modern weak formulations, and addresses diverse questions: extinction, global behavior, stability, regularity, singularity formation, and flow in generalized or singular geometries.

1. Classical Properties and Foundational Theorems

The planar (Euclidean) CSF for embedded closed curves is characterized by several fundamental results. Gage–Hamilton established that any embedded convex curve remains convex and contracts homothetically to a round point in finite time; the solution shrinks to extinction while approaching a circle under rescaling. Grayson extended this by proving that any embedded closed curve in $\mathbb{R}^2$, regardless of convexity, becomes convex in finite time and then follows the Gage–Hamilton scenario.

Formally, with $X : S^1 \to \mathbb{R}^2$ parameterized by arclength, the evolution is

$\partial_t X = \partial_{ss} X,$

where $s$ is arclength. Notable monotonicity properties hold:

• Total absolute curvature $\int |k| ds$ is non-increasing.
• Length $\partial_t X = kN$0 is non-increasing and decays at least exponentially compared to a round circle.
• Enclosed area decreases at rate $\partial_t X = kN$1; extinction occurs when enclosed area vanishes.

These results generalize to certain nonplanar and non-Euclidean settings, as described below (Guidotti, 2023).

2. Weak Solutions and Geometric Singularities

While the flow is classically smooth until first singularity (blow-up of curvature or vanishing length), various generalized solutions exist to extend past singularities, including:

• Level-set and viscosity solutions, capable of passing through non-embedded or non-smooth configurations (Dobbins, 2021). Level-set flows are constructed as limits using barrier arguments or as the evolution of a distance function under a degenerate parabolic PDE.
• Brakke flows: measure-theoretic generalizations, permitting topology change but sacrificing uniqueness.
• Semi-discrete time-approximation: The global construction of solutions for CSF with finite total absolute curvature involves evolving the piecewise linearized curve by alternating linear heat flow and reparameterization. This yields a globally defined geometric flow even in the presence of finitely many singularities (cusps due to loop-shedding), with controlled geometric energies and strong compactness properties. Singularities correspond to events where the tangent vanishes (odd-order zero), but the construction ensures global existence in the BV-class and recovers a classical solution away from isolated singular times (Guidotti, 2023).

An explicit characterization: | Approach | Regularity Across Singularities | Uniqueness | Physical/geometric model | |-------------------------|:------------------------------:|-------------|-----------------------------------| | Classical CSF | Breaks at singularities | Yes—until singularity | Smooth curves | | Level-set/viscosity | Exists globally | Yes | Weak/metric spaces, non-smooth | | Semi-discrete | Exists globally, finite jumps | Yes (where classical unique) | BV curves w/ finite corners | | Brakke flow | Exists globally | Generally no| Varifold, geometric measure theory|

3. High Codimension and Non-Euclidean Settings

a. Higher Codimension and Convex Projections

For curves in $\partial_t X = kN$2, CSF is given by $\partial_t X = kN$3, with $\partial_t X = kN$4, $\partial_t X = kN$5 taking values in the normal bundle. Embeddedness is not generally preserved. However, if the curve admits a convex (and one-to-one) projection onto a 2-plane, strong regularity and extinction results are recovered:

• The planar projection remains convex and smooth until shrunk to a point.
• The singularity is always of Type I (curvature blows up at most at the rate $\partial_t X = kN$6), and after rescaling, the flow converges $\partial_t X = kN$7 to a round circle of multiplicity one in a 2-plane; type-II (translator) singularities are excluded (Sun, 16 Oct 2025, Sun, 2024).

A key analytical tool is Huisken's entropy functional, which controls possible singularity models and guarantees the flow avoids non-circular blow-up limits in low-entropy regimes (Litzinger, 2023). Furthermore, any closed immersed curve in $\partial_t X = kN$8 can be perturbed in $\partial_t X = kN$9 so that its projection is convex, hence its flow shrinks roundly (Sun, 16 Oct 2025, Sun, 2024).

b. CSF in Non-Euclidean and Singular Geometries

CSF is well-posed on Riemannian surfaces, including those with:

• Conical singularities: The evolution near a cone point is governed by a degenerate quasilinear parabolic PDE with cone-differential operator coefficients. Short-time existence, uniqueness, and precise curvature asymptotics are established, with the evolving curve fixed at cone singularities. CSF can yield finite-time collapse or convergence to a geodesic, echoing classical results (Roidos et al., 2020).
• Non-convex-at-infinity surfaces: For warped products with strictly increasing warp function (so compactness at infinity fails), embedded curves become 'graphs' over the $k$0-factor in finite time. The graph property is preserved (i.e., the curve never becomes vertical), yielding global existence and dynamic control via angle-function evolution equations (Fujihara, 2024).
• Metric-affine planes: CSF extends to metrics with affine connections (with parallel, bounded contorsion) by adding a correction to the curvature and ensuring convexity/vanishing area is preserved, resulting in extinction to a round point (Rovenski, 2020).

4. Boundary Value Problems and Coupled Systems

a. Dirichlet and Free Boundary Problems

CSF for open arcs with fixed (Dirichlet) endpoints on convex or constant-curvature backgrounds remains embedded, with barrier arguments and distance comparison methods ensuring convergence to the unique geodesic between endpoints (Allen et al., 2012). For arcs meeting a fixed convex barrier orthogonally (free boundary), as in high codimension, a refined maximum principle at the boundary yields sharp curvature and derivative estimates up to the boundary. Under an entropy bound $k$1, the only possible singularities are modeled on shrinking semicircles, or else the flow converges smoothly to the unique orthogonal chord (Nguyen et al., 24 Feb 2026).

b. Area-Preserving and Forced Flows

The area-preserving CSF, with a Lagrange multiplier to maintain constant enclosed area, and Neumann free boundary conditions outside a convex domain, yields a flow where no singularity develops under suitable geometric constraints, and the curve subconverges to an arc of a circle meeting the boundary at right angles (Mäder-Baumdicker, 2015).

c. Coupled Evolution and Diffusion

Numerical and analytical frameworks have been investigated for CSF coupled to parabolic field equations (diffusion on the curve), leading to semi-discrete finite element schemes with optimal convergence properties. The geometric evolution is robust to lower-order normal forcing and parabolic transport on the moving domain (Pozzi et al., 2015).

5. Analytical and Numerical Methodologies

A persistent theme is the combination of maximum principle, monotonicity identities (e.g., Huisken’s entropy), and barrier arguments for regularity and singularity analysis. Recent advances provide:

• Variational characterizations: Interpreting CSF as tangent-aligning gradient flow allows energy methods for infinite-length or noncompact curves, establishing convergence to straight lines under finite “direction energy” (Miura et al., 4 Apr 2025).
• Spectrally accurate numerical schemes: The semi-discrete scheme for CSF with finite total absolute curvature (heat-flow plus reparameterization) is unconditionally stable, handles singularities via loop-shedding events, and converges to the continuous flow (Guidotti, 2023).
• Localized boundary estimates: Stahl-type localized maximum principles adapted for free-boundary CSF yield dilation-invariant curvature and higher-derivative control right up to the boundary, critical in high codimension and under geometric constraints (Nguyen et al., 24 Feb 2026).

6. Applications and Extensions

CSF underpins the smooth deformation theory of two-dimensional projective planes, deformation retraction in combinatorial topology, and geometric optimality in shape evolution. On $k$2, continuous dependence in Fréchet distance for weak (level-set) flows with area-bisecting initial data is crucial for applications to pseudocircle arrangements and oriented matroids (Dobbins, 2021, Hsu, 2013).

Current research directions include:

• Quantitative estimates for convergence and singularity profiles.
• Extension of the semi-discrete framework to higher codimension, surface flows, and non-Euclidean settings.
• Classification of generic and non-generic singularity models in both low and high codimension.
• Development of unified variational frameworks for curve/surface flows, including elastic and diffusion flows.

These advances collectively demonstrate the robust and versatile analytical structure of the curve shortening flow, and its centrality to modern geometric analysis and applied PDE theory.