Finite-orbit variational rigidity characterises homothetic ellipses, with quantitative obstructions for all other convex pairs. The self-Minkowski billiard integrability is resolved without symmetry or positive-curvature assumptions.
Geometric Analysis
Solved question 2 from arXiv:2602.12896 affirmatively.
The task is to classify smooth positively curved pairs of planar convex curves for which every finite physical orbit of the alternating symplectic billiard has non-positive fixed-endpoint second variation. Such pairs are the independently translated homothetic ellipses, and all of their nontrivial finite Hessians are negative definite. The proof combines an exact finite-block averaging identity with a common area-preserving normalization and independent scaling. The same identity bounds the orbit length at which a positive second variation occurs outside the classified family. An exact polar reduction and a Schur-complement identity transfer the theorem to Minkowski billiards, giving a classification of totally integrable metric–table pairs for strongly convex gauges. In the self-Minkowski case, total integrability forces the gauge to come from an inner product. No symmetry or positive-curvature hypothesis is imposed in this case and curvature is obtained from calibration of continuous invariant graphs.
Ellipse: table and gauge become a round disk together under a linear change of coordinates.