Geometry-Constrained Diffusion and Shape Contraction on Convex Curves

We study a diffusion process defined on smooth,strictly convex closed curves and constrained by the geo-metric conditions that characterize admissible curvatureprofiles. The macroscopic geometry is described througha normalized radius-of-curvature profile evolving under aprescribed curvature-dependent inward flow. On a fixedreference curve, Brownian motion with respect to ar-clength induces a geometry-dependent diffusion gener-ator. Since unconstrained diffusion does not, in general,preserve the normalization and closure conditions of thegeometric profile, we construct an arclength-weighted or-thogonal projection onto the tangent space of the admis-sible shape manifold.For a smooth positive reference profile, the resultingconstrained diffusion preserves the geometric momentconstraints and generates a dissipative evolution in thecorresponding weighted L2 space. We establish well-posedness of the frozen-geometry evolution and short-time preservation of positivity. Our main result gives anexplicit contraction estimate for a convex-combinationupdate of the macroscopic profile. In particular, the con-traction rate is bounded below in terms of the diffusioncoefficient, the reference scale, and the maximum of thereference profile. A circular benchmark provides an exactFourier-mode characterization of the contraction mecha-nism. The results establish a rigorous single-interventionframework for geometry-constrained diffusion on convexcurves