Shape selection in non-Euclidean plates

TL;DR

This paper studies the geometry and energy minimization of negatively curved surfaces embedded in three-dimensional space, revealing existence, singularity bounds, and constructing low-energy piecewise smooth configurations with periodic profiles.

Contribution

It establishes the existence of large smooth isometric immersions, derives curvature bounds, and constructs low-energy piecewise smooth immersions with periodic structures.

Findings

Existence of smooth immersions of large hyperbolic disks into R^3.

Lower bounds on principal curvatures for smooth immersions.

Construction of low-energy, piecewise smooth, periodic immersions.

Abstract

We investigate isometric immersions of disks with constant negative curvature into $\mathbb{R}^3$, and the minimizers for the bending energy, i.e. the $L^2$ norm of the principal curvatures over the class of $W^{2,2}$ isometric immersions. We show the existence of smooth immersions of arbitrarily large geodesic balls in $\mathbb{H}^2$ into $\mathbb{R}^3$. In elucidating the connection between these immersions and the non-existence/singularity results of Hilbert and Amsler, we obtain a lower bound for the $L^\infty$ norm of the principal curvatures for such smooth isometric immersions. We also construct piecewise smooth isometric immersions that have a periodic profile, are globally $W^{2,2}$, and have a lower bending energy than their smooth counterparts. The number of periods in these configurations is set by the condition that the principal curvatures of the surface remain finite and grows approximately exponentially with the radius of the disc. We discuss the implications of our results on recent experiments on the mechanics of non-Euclidean plates.