Length-constrained curve diffusion flow for open curves with endpoints on two intersecting lines

Abstract

We study the curve diffusion flow for open planar curves whose endpoints are constrained to lie on two fixed straight lines that intersect at an angle $θ(\in(0,π)) $. For every such angle, we prove that under suitable initial conditions the flow exists globally in time. Moreover, we show that the evolving curve converges - exponentially and in the smooth topology - to the circular arc of a sector whose central angle is exactly $θ$ and whose arc length equals that of the initial curve. This result reveals how a length-preserving fourth-order geometric flow can straighten out a curve's shape while respecting boundary constraints, ultimately driving it toward a unique equilibrium: the circular arc spanning the prescribed angle. This provides a complete description of the long-time behaviour of this fourth-order geometric flow with mixed boundary conditions.