Interior caustics and rigidity near noncircular ellipses

Abstract

We prove local rigidity near every fixed noncircular ellipse from integrable rational caustics in an arbitrary nonempty open rotation interval $I\subset(0,1/2)$. For each fixed $r>11/2$ and bound on the $H^r$ norm of the elliptic graph displacement, every sufficiently $C^1$-small positively curved perturbation admitting such caustics at all rational rotations in $I$ is an ellipse. A fixed $C^6$ bound may replace the Sobolev bound, with $C^1$ smallness retained. The caustic hypothesis can be restricted to a suitably chosen sequence of rational rotations in $I$ with any prescribed rational $ρ_*\in I$ as its unique accumulation point, together with finitely many supplementary rotations in $I$; this family depends only on the reference ellipse, interval, and anchor. The main linear result classifies all real $L^2$ displacements satisfying the constant rational orbit-sum equations throughout $I$ as infinitesimal ellipse deformations, and establishes their analyticity and rational continuation in the complexified boundary parameter. The proof combines analytic continuation and elimination of monodromy with quantitative caustic estimates, finite detection, and Fourier interpolation.