FOLIATED PLATEAU PROBLEMS, GEOMETRIC RIGIDITY AND EQUIDISTRIBUTION OF CLOSED k-SURFACES

The paper states and sketches a correct proof of the rigidity of the hyperbolic marked area spectrum for k-surfaces (Theorem 4.5: MAS_{k,h} = MAS_{k,h0} iff h ≅ h0) and explains the key mechanism: equality of areas forces sect_h = −1 along the tangent planes of all quasi-Fuchsian k-surfaces (via Gauss–Bonnet/Gauss equation), and an equidistribution argument à la Kahn–Marković yields a limiting laminar measure with full support on T^1X, so continuity implies sect_h ≡ −1 everywhere and Mostow then gives isometry . By contrast, the model’s Step 4 replaces the equidistribution step with an unsubstantiated topological density claim (“tangent planes of closed k-surfaces are dense in the unit tangent bundle”). The density result the model alludes to is about compact leaves being dense in the lamination of pointed k-surfaces Sk(M) (Labourie), not a statement that the union of tangent planes to closed k-surfaces is dense in T^1M; the paper does not use nor establish that stronger claim, and instead relies on a full-support limiting measure from a Kahn–Marković sequence to conclude sect_h ≡ −1 . Hence the model’s argument is incomplete at the decisive step, while the paper’s argument is correct.