Problems in surface theory/ membranes/hypersurface singularities

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AbstractAward: DMS-0071729Principal Investigator: Brian SmythThis proposal is predominantly concerned with qualitative aspectsof the theory of surfaces as well as its application to theconstruction of a complete theory of equilibria for the generalelastic membrane. The work on surface theory is uncoveringpreviously unsuspected qualitative behavior of the principalfoliations near an isolated singularity (umbilic) in a smoothsurface in space, which strikingly distinguishes them fromgeneral foliations. This behavior explains the Caratheodoryconjecture and, in many cases, proves the conjecture as a merenumerical consequence. The work on elastic membranes establishesthe equilibrium equation of the most general stretched elasticmembrane in equilibrium under an applied force field --- afar-reaching generalization of the classical Young-Laplaceequation which was only valid for isotropic membranes. Thecontext for this work is the unit circle bundle of the membranesurface equipped with the canonical lift (Sasaki metric) of theinduced metric on the membrane surface; the key idea is theconstruction of a vector-valued 1-form on this bundle whichcompletely encodes the responding forces of surface tension setup within the membrane.Surfaces are mathematical entities which are fundamental to theunderstanding of the structure of matter in physics, chemistryand biology. The surface at the interface of two different typesof matter is frequently modeled by an elastic membrane. Anobserver of an elastic membrane in equilibrium can see the shapeof the membrane, understand the forces that are being applied andeven measure the average surface tension but has no idea of theresponding forces of surface tension set up within themembrane. This work reveals the precise relationship betweenthese observables and the internal surface tension forces for thegeneral membrane. It follows that the response is never entirelypredictable from the observables (membrane geometry, appliedforce, average surface tension) but that a certain core behavioris determined --- except for a very remarkable family ofmembranes (whose exceptional nature is defined solely by theirgeometry and not their physics) which live by a different set ofrules. By example, it is shown that the common ad hoc assumptionsof isotropy are seriously flawed. The authors' work on theprincipal foliations of surfaces has already provided a keyresult which has been applied by Rozoy to settle a long-standingproblem on the symmetry of a liquid ball in general relativity,as well as relating streamlines of a plane fluid flow to surfacetheory.