Together with Eliot Fried, I was able to confirm the results of Cox and Jones by analyzing an area minimization problem with the boundary conditions determined by the geometry described above [ArXiv]. Besides considering the case in which the two line segments are parallel, we investigated the case when they are not parallel. For this configuration, one can analyze the stability of certain helices. We found that (i) for sufficiently long, thin cylinders, no helicoid is stable, (ii) regardless of the dimensions of the cylinder, no helicoid that contains a little over half of a rotation is stable, and (iii) as long as the cylinder is not overly long and thin, there is a non flat stable helicoid.

Since the procedure we used gave explicit integral representations of the bending moduli in terms of the interaction potential, once a particular potential is chosen, it is possible to obtain values for the moduli [ArXiv]. Upon chosing a Gaussian potential inspired by the works of Berne and Pechuckas and Gay and Berne, Eliot and I computed the bending moduli, which were found to depend on two parameters associated with the potential. Adjusting these parameters yields values for the moduli matching most values reported in the literature. A result of note is our computations yielded a negative value for the modulus associated with the Gaussian curvature.

Together with Denis Hinz, Eliot and I wrote a paper that presented the ideas and results related to our work on the generalized transport theorem in a way that makes them accessible to researchers working in the field of continuum physics. This was achieved in part by considering three concrete, illustrative examples in which the terms appearing in the generalized transport theorem were calculated numerically. Eliot and I's desire to establish the transport theorem was motivated by the existence of evolving irregular domains that occur in nature, but the proof is a work in pure mathematics. Thus, many researchers who could potentially benefit from our work might not become exposed to it and, so, presenting our results specifically to this audience could prove useful.

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