Hyperbolic Monge-Ampere equations and applications to thin sheet elasticity

I will talk about some geometric questions that arise in the study of soft/thin objects with negative curvature. I will discuss some recent results on existence/non-existence of solutions to hyperbolic Monge-Ampere equations with various degrees of regularity, with an emphasis on numerical methods for constructing "rough" solutions. I will then discuss some applications of our results to (i) the occurrence of "geometric" defects that are invisible to the energy, but play a crucial role in determining the global morphology, (ii) a generalization of the Sine-Gordon equation to describe "rough" hyperbolic surfaces with constant negative curvature, and (iii) the important role of regularity in quantitative versions of the Hilbert-Efimov theorem on the nonexistence of C^2 isometric immersions of the Hyperbolic plane into R^3, and (iv) studying the mechanics of leaves, flowers, and sea-slugs.

This is joint work with Toby Shearman and Ken Yamamoto.