Constant Gaussian Curvature:

In references [1-4] we studied the problem of constructing and analyzing the bending energy for isometric immersions of \(\mathbb{H}^2\) into \(\mathbb{R}^3\). We were motivated by experiments on hydrogel disks with programmed constant Gaussian curvature. With decreasing thickness these hydrogels adopted a saddle shape that transitioned to multi-wave shapes with decreasing thickness; see the figure to the below left. Motivated by these experimental results, in [1] we constructed \(W^{2,2}\) isometric immersions of \(\mathbb{H}^2\) with periodic profiles that qualitatively these experimentally realized shapes; see the lower right figure. We showed that for purely geometrical reasons, the number of waves in these isometric immersions must refine with increasing radius matching observations in crochet models of \(\mathbb{H}^2\). Indeed, we provided strong numerical evidence that for large radii these periodic shapes are energetically preferred over saddle shapes; see Fig. 1. However, in [4] we rigorously proved by finding ansatz free scaling laws that for the range of parameters used in these experiments the global minimizers of the variational problem are the saddle shapes for all thickness values.

In [1] we show that the self-similar shapes observed in torn plastic can be explained by low energy piecewise smooth isometric immersions.

More information on my work in this area can be found in the publications listed below.

Video of a presentation given at the IMA

Publications Related to this Work:

- Gemmer, J. A., & Venkataramani, S. C. (2011). Shape selection in non-Euclidean plates. Physica D: Nonlinear Phenomena, 240(19), 1536-1552.
- Gemmer, J. (2012).
*Shape Selection in the non-Euclidean Model of Elasticity*(Doctoral dissertation, The University Of Arizona). - Gemmer, J. A., & Venkataramani, S. C. (2012). Defects and boundary layers in non-Euclidean plates. Nonlinearity, 25(12), 3553.
- Gemmer, J. A., & Venkataramani, S. C. (2013). Shape transitions in hyperbolic non-Euclidean plates. Soft Matter, 9(34), 8151-8161
- Gemmer, J. A., Venkataramani, S. C., Sharon, E. (2016) Isometric immersions and self-similar buckling in Non-Euclidean elastic sheets. (preprint).

Potential Graduate Student Research:

So far my work has focused on studying swelling thin elastic sheets
with a specific swelling pattern, namely one whose geometry corresponds
to constant negative Gaussian curvature. I believe that the results I
have in the case of constant Gaussian curvature will still hold for
more generic swelling patterns. One
way a student could assist me is by designing, running,
and analyzing numerical simulations of swelling thin elastic sheets
with generic radially symmetric swelling patterns. This
project
could be undertaken by a student with experience in differential
equations, numerical analysis, and programming in a higher level
language like Mathematica
of Matlab. The essential question the student will be exploring is the
role of geometry in determining complex morphologies observed in leaves
and torn plastic. I believe that this project could nucleate into
further research that would have a high impact on the field.

**Another project that could be pursued by a graduate student is to develop and analyze a mathematical of swelling thin elastic sheets that incorporates dynamic and stochastic effects. **My past work has shown that local but not global minimum seem to be what is observed in practice. This
project
could be undertaken by a student with experience in differential
equations, numerical analysis, elasticity, calculus of variations and programming in a higher level
language like Mathematica
of Matlab. This research could serve as the beginning of a Ph.D. dissertation.