Rigidity and Nonrigidity of Smooth Tight Surfaces

We study global isometric rigidity of smooth embedded tight surfaces in Euclidean three-space when zero is a regular value of the Gaussian curvature. We construct noncongruent isometric embeddings in every positive genus. In the same range of genera, we construct globally rigid regular-tight embeddings whose negative-curvature regions contain open annuli foliated by closed asymptotic curves. We also prove that the embeddings for which all closed asymptotic curves are hyperbolic form a dense $G_\delta$ in the relative smooth topology of the regular-tight class; these embeddings are globally rigid by the theorem of Han and Khuri. The two constructions use a common exact finite-frequency realization of neutral annuli. A nonvanishing scalar Schur obstruction gives finite uniqueness across a neutral band, whereas a suitable zero of that obstruction supplies a ruled replacement carrying compactly supported infinitesimal bendings. A final polarization produces the noncongruent isometric pairs. The genericity theorem follows from return-map transversality realized by tightness-preserving support deformations.