Branch points and non-density for finite-bending isometric immersions of hyperbolic surfaces

The space of $W^{2,2}$-isometric immersions of a surface into ${\mathbb{R}}^3$ arises naturally in the variational theory of thin elastic sheets: it is precisely the finite-bending class, where the bending energy — the $L^2$-norm of the second fundamental form — is finite. For sheets with negative Gaussian curvature, previous work has identified branch points, where “too many” asymptotic directions meet — or, equivalently, where the index of the Gauss map is not $-1$ — as a potentially important mechanism in shape selection and pattern formation. Such branch points are precluded for $C^2$-isometric immersion. We show that this index-based notion of branch points extends to the full finite-bending class: Namely, for every $W^{2,2}$ isometric immersion of a negatively-curved surface, the index of the Gauss map is well-defined at every point, and the set of branch points is discrete. We further show that the index is stable under $W^{2,2}$-convergence, and thus, an isometric immersion with branch points cannot be approximated by $C^2$-isometric immersions. Conversely, we show that every negatively-curved metric locally admits $W^{2,2}$-isometric immersions (in fact, $C^{1,1}$) with branch points of arbitrary order. Consequently, $C^2$-isometric immersions are, in general, not dense among finite-bending ones, in stark contrast with the flat and positively curved cases.