The one-dimensional instance of mean curvature flow for embedded planar curves in which each point moves in the inward normal direction at speed equal to the scalar curvature (signed curvature), causing length decrease and often convexification; commonly abbreviated CSF.

Principle

CSF is the gradient flow of curve length: points move to decrease length locally, so curvature-driven motion smooths small-scale features and tends to round convex curves while preserving embeddedness until extinction or singularity.

Demonstration

A round planar circle shrinks homothetically to a point under CSF; any smooth embedded closed planar curve becomes convex in finite time and then collapses to a round point, by the Gage–Hamilton–Grayson phenomenon in the plane.

Misapplication

Assuming results for embedded closed curves in the plane extend unchanged to open curves, non-planar curves, or higher-codimension curves without checking for boundary conditions or new instability mechanisms.

Consequence

Correct use yields monotone length decrease, smoothing and eventual extinction behavior for closed embedded curves, and specific curvature bounds; it provides canonical geometric simplification for planar curves.

Reversal

An outward curvature-driven expansion (inverse curvature flow) can enlarge curves or create singular outward motion; time reversal of CSF is typically ill-posed and increases length rapidly.

Boundary

Formulated for curves in the plane (or surfaces of codimension one) and for closed embedded curves; excludes fourth-order flows like curve diffusion, flows with anisotropic speed laws, or maps between manifolds without further structure.

Semantic Tension

Tension arises between CSF and higher-order geometric flows (curve diffusion) or extrinsic flows in higher codimension: CSF is second-order and strongly tied to planar embeddedness, while other flows preserve different invariants or regularity.

Synthesis

Curve shortening flow is the planar, one-dimensional mean curvature evolution that decreases curve length by moving points along normals at speed equal to curvature, producing convexification and canonical collapse for closed embedded curves.