For a
closed material membrane in equilibrium in a force field we investigate whether the external observables (membrane geometry and force field) determine the internal membrane response (the stress tensor
T), when the mean stress
Tr
T is known. For membranes with boundary the indeterminacy of the response is classical. For closed membranes the geometry decides the question. We show uniqueness for all but a class of soliton surfaces — the
globally isothermic surfaces (§1, 2); the physical phenomenon exhibited by any closed globally isothermic membrane in equilibrium is that, with all observables static, there is in total a 1-parameter family of responses with the same mean stress — all canonically determined by membrane geometry (Theorem 1). There exist closed embedded globally isothermic surfaces of every genus. The recognition of the role of these soliton surfaces settles the old classification problem for the space of static shears in any closed membrane, explicitly identifying all static shears, where before only a genus-dependent dimension bound was known (Theorem 2).
Communicated by G.W. Gibbons
Partially supported by NSF Grant DMS00 71729Acknowledgement. I am grateful to the referee for pointing out that Theorem 1 resonates with a recent characterization of isothermic surfaces [5], namely that they are precisely those surfaces which are not determined up to conformal diffeomorphism by the trace-free part of the second fundamental form (see also [11]).
This work was begun at the Max Planck Institute for Mathematics in the Sciences in Leipzig where I was a visitor in 2000 and continued at the Max-Planck-Institut f
ur Mathematik in Bonn in 2001. It is a pleasure to record my thanks to both institutes. I thank Robert Finn for conversations on this work when we were both visitors in Leipzig and Chuu-Lian Terng for pointing me toward the lecture notes of Burstall [4] on isothermic surfaces.