Target Flow in Planar Curves

• Target Flow is a curvature-driven evolution that deforms an initial planar curve smoothly to a fixed target using an explicit ambient forcing field.
• The methodology employs a force derived from target curvature in a maximal tubular neighborhood to correct both position and curvature, ensuring exponential convergence.
• The analytical framework guarantees global well-posedness and exponential decay of deviations using quasilinear parabolic PDE techniques and normal graph assumptions.

A "target flow" is a geometric evolution law for planar curves designed to deform a given initial (source) curve smoothly to a fixed embedded target curve via a curvature flow with an explicit ambient forcing term. This framework generalizes traditional curve shortening and related mean curvature flows, enabling convergence to non-convex, arbitrary embedded targets without requiring convexity or additional length/area constraints. The construction—introduced by Cuthbertson, Wheeler, and Wheeler—resolves longstanding problems in geometric analysis regarding the flow of one embedded curve to another, notably extending previous results that were restricted to convex targets or required complicated rescalings. The target flow admits a rigorous analysis within the theory of quasilinear parabolic PDEs with degenerate highest-order terms.

1. Definition and Evolution Equation

The target flow is formulated for an embedded target curve $\eta:S^1\to\mathbb R^2$ parameterized by arclength $u\mapsto\eta(u)$. An evolving curve $X(u,t)=\gamma(u,t)\subset\mathbb R^2$ follows the PDE: $$\partial_{t}X = \kappa\,\nu + V^{\eta}(X) = (k + F)\nu,$$ where:

• $\kappa = k\nu$ is the curvature vector field of $X(\cdot,t)$, with scalar curvature $k(u,t)$ and unit inward normal $\nu(u,t)$,
• $V^\eta:\mathbb R^2\to\mathbb R^2$ is an explicit ambient vector field determined solely by $\eta$,
• $u\mapsto\eta(u)$0 is the normal component of $u\mapsto\eta(u)$1 at $u\mapsto\eta(u)$2.

This evolution unifies curvature-driven contraction with a field that “pulls” the curve toward $u\mapsto\eta(u)$3 both in position and in curvature.

2. Structure and Interpretation of the Ambient Forcing Field

The defining feature of the target flow is the construction of the ambient field $u\mapsto\eta(u)$4, which acts in the maximal tubular neighborhood $u\mapsto\eta(u)$5 of $u\mapsto\eta(u)$6. In normal coordinates,

$u\mapsto\eta(u)$7

where $u\mapsto\eta(u)$8 is the inward unit normal of $u\mapsto\eta(u)$9 and $X(u,t)=\gamma(u,t)\subset\mathbb R^2$0 is a signed normal offset, the field is prescribed as: $X(u,t)=\gamma(u,t)\subset\mathbb R^2$1 with $X(u,t)=\gamma(u,t)\subset\mathbb R^2$2 the curvature of $X(u,t)=\gamma(u,t)\subset\mathbb R^2$3 and $X(u,t)=\gamma(u,t)\subset\mathbb R^2$4 a sufficiently large constant.

• The term $X(u,t)=\gamma(u,t)\subset\mathbb R^2$5 ensures the limiting curvature of the evolving curve matches $X(u,t)=\gamma(u,t)\subset\mathbb R^2$6.
• The term $X(u,t)=\gamma(u,t)\subset\mathbb R^2$7 induces exponential attraction to $X(u,t)=\gamma(u,t)\subset\mathbb R^2$8 along the inward normal. For initial data that is a constant normal shift of $X(u,t)=\gamma(u,t)\subset\mathbb R^2$9, $$\partial_{t}X = \kappa\,\nu + V^{\eta}(X) = (k + F)\nu,$$0 decays exponentially: $$\partial_{t}X = \kappa\,\nu + V^{\eta}(X) = (k + F)\nu,$$1.

This construction guarantees that the trajectory “locks onto” the target manifold in both position and curvature as $$\partial_{t}X = \kappa\,\nu + V^{\eta}(X) = (k + F)\nu,$$2.

3. Admissible Initial Data: Uniformly Normal Graphs

The flow is globally well-posed within the class of curves that are uniformly normal graphs over $$\partial_{t}X = \kappa\,\nu + V^{\eta}(X) = (k + F)\nu,$$3:

1. The initial curve $$\partial_{t}X = \kappa\,\nu + V^{\eta}(X) = (k + F)\nu,$$4 lies entirely in $$\partial_{t}X = \kappa\,\nu + V^{\eta}(X) = (k + F)\nu,$$5.
2. There exists $$\partial_{t}X = \kappa\,\nu + V^{\eta}(X) = (k + F)\nu,$$6 and a representation $$\partial_{t}X = \kappa\,\nu + V^{\eta}(X) = (k + F)\nu,$$7.
3. $$\partial_{t}X = \kappa\,\nu + V^{\eta}(X) = (k + F)\nu,$$8 belongs to the little Hölder space $$\partial_{t}X = \kappa\,\nu + V^{\eta}(X) = (k + F)\nu,$$9 for $\kappa = k\nu$0.

The maximal strip $\kappa = k\nu$1 is: $\kappa = k\nu$2 where $\kappa = k\nu$3, $\kappa = k\nu$4 are the extremal values of $\kappa = k\nu$5.

4. Main Convergence Theorem

For any smooth (or analytic) embedded $\kappa = k\nu$6 and initial data $\kappa = k\nu$7 in the uniformly normal graphical class, there exists a unique global solution $\kappa = k\nu$8 to the target flow that obeys:

• $\kappa = k\nu$9 remains a normal graph over $X(\cdot,t)$0 with $X(\cdot,t)$1,
• For $X(\cdot,t)$2, $X(\cdot,t)$3 is smooth (or analytic if $X(\cdot,t)$4 is analytic),
• $X(\cdot,t)$5 exponentially in every $X(\cdot,t)$6 norm as $X(\cdot,t)$7, and thus $X(\cdot,t)$8 smoothly.

Explicitly, for constants $X(\cdot,t)$9,

$k(u,t)$0

ensuring exponential convergence of the evolving curve to the target in all derivatives (Cuthbertson et al., 2024).

5. Analytical Proof Strategy

The existence, uniqueness, and exponential convergence are established through several analytic layers:

• The evolution is reformulated as a quasi-linear parabolic PDE for the graph function $k(u,t)$1.
• A barrier comparison with explicit ODE exponential decay solutions yields $k(u,t)$2 bounds.
• Gradient and higher derivative estimates for $k(u,t)$3 are attained via maximum principles and bootstrapping arguments.
• Existence and uniqueness derive from the sectorial property of the linearized operator in little Hölder spaces, utilizing Angenent’s theory for nonlinear degenerate parabolic equations.
• The precise long-term exponential decay of higher derivatives is achieved via refined maximum principles and Schauder-type estimates.

6. Illustrative Examples and Behaviors

Circle Target

For $k(u,t)$4, with $k(u,t)$5, the ambient field reduces to: $k(u,t)$6 producing a radially symmetric force that intensifies as $k(u,t)$7.

Non-convex ("Bean") Target

For non-convex $k(u,t)$8, $k(u,t)$9 remains purely normal but its magnitude varies; at regions of high curvature, the $\nu(u,t)$0 term is substantial, particularly on the inner side of loops. This spatial specificity of the field corrects both positional and curvature deviations, even in the absence of symmetry or convexity.

7. Extensions, Open Questions, and Context

• Higher Co-dimension: Whether $\nu(u,t)$1 can be generalized to force hypersurfaces in $\nu(u,t)$2 to an embedded target remains open.
• Graphical Assumption Relaxation: Whether convergence holds for initial data not globally a normal graph, or under finite normal turning, is unresolved.
• Regularity Thresholds: Replacing the little Hölder space $\nu(u,t)$3 with weaker regularity spaces (e.g., $\nu(u,t)$4 or Sobolev) is a topic for further analysis.
• Higher-dimensional Generalizations: The approach lays groundwork toward the solution of Yau’s embedding problem for submanifolds of arbitrary codimension.

Earlier flows for the convex-to-convex curve matching problem required auxiliary rescalings (e.g., length- or area-preservation) as in Lin–Tsai, Gage–Li, Chou–Zhu, and Gao–Zhang. The target flow fundamentally extends these results to general embedded curves by leveraging an explicit ambient forcing term rather than intrinsic rescalings (Cuthbertson et al., 2024).

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The target flow thus provides a robust, analytically validated mechanism for deforming curves in the plane to arbitrary embedded targets, with precise control on the geometry of the evolving curve and minimal restrictions on the target, representing a substantial advancement in geometric evolution theory.