Problems in surface theory/ membranes/hypersurface singularities

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

Abstract Award: DMS-0071729 Principal Investigator: Brian Smyth This proposal is predominantly concerned with qualitative aspects of the theory of surfaces as well as its application to the construction of a complete theory of equilibria for the general elastic membrane. The work on surface theory is uncovering previously unsuspected qualitative behavior of the principal foliations near an isolated singularity (umbilic) in a smooth surface in space, which strikingly distinguishes them from general foliations. This behavior explains the Caratheodory conjecture and, in many cases, proves the conjecture as a mere numerical consequence. The work on elastic membranes establishes the equilibrium equation of the most general stretched elastic membrane in equilibrium under an applied force field --- a far-reaching generalization of the classical Young-Laplace equation which was only valid for isotropic membranes. The context for this work is the unit circle bundle of the membrane surface equipped with the canonical lift (Sasaki metric) of the induced metric on the membrane surface; the key idea is the construction of a vector-valued 1-form on this bundle which completely encodes the responding forces of surface tension set up within the membrane. Surfaces are mathematical entities which are fundamental to the understanding of the structure of matter in physics, chemistry and biology. The surface at the interface of two different types of matter is frequently modeled by an elastic membrane. An observer of an elastic membrane in equilibrium can see the shape of the membrane, understand the forces that are being applied and even measure the average surface tension but has no idea of the responding forces of surface tension set up within the membrane. This work reveals the precise relationship between these observables and the internal surface tension forces for the general membrane. It follows that the response is never entirely predictable from the observables (membrane geometry, applied force, average surface tension) but that a certain core behavior is determined --- except for a very remarkable family of membranes (whose exceptional nature is defined solely by their geometry and not their physics) which live by a different set of rules. By example, it is shown that the common ad hoc assumptions of isotropy are seriously flawed. The authors' work on the principal foliations of surfaces has already provided a key result which has been applied by Rozoy to settle a long-standing problem on the symmetry of a liquid ball in general relativity, as well as relating streamlines of a plane fluid flow to surface theory.