Problems in surface theory/ membranes/hypersurface singularities

AbstractAward：DMS-0071729首席研究员：Brian Smith该提案主要涉及曲面理论的定性方面，以及其在构建广义弹性膜平衡的完整理论中的应用。 表面理论的工作揭示了空间光滑表面上孤立奇点（脐点）附近的主叶理的先前未被怀疑的定性行为，这使它们与一般叶理有了显著的区别。这种行为解释了Caratheodory猜想，并在许多情况下，证明猜想作为merenumerical后果。对弹性膜的研究建立了在外力场作用下处于平衡状态的最一般的拉伸弹性膜的平衡方程--经典的仅对各向同性膜有效的Young-Laplace方程的深远推广。本文的背景是具有正则升力的膜表面的单位圆丛（Sasaki度规）的性质;关键思想是在这个丛上构造一个向量值1-形式，它完全编码了膜内表面张力设置的响应力。表面是理解物理学中物质结构的基本数学实体，chemistry化学and biology生物.在两种不同类型物质的界面处的表面通常用弹性膜来模拟。一个处于平衡状态的弹性膜的观察者可以看到膜的形状，了解施加的力，甚至测量平均表面张力，但不知道膜内建立的表面张力的响应力。这项工作揭示了这些观测值和一般膜的内部表面张力之间的精确关系。 由此可见，从可观测的（膜的几何形状、作用力、平均表面张力）中，反应永远无法完全预测，但某些核心行为是确定的--除了一个非常引人注目的膜家族（其特殊性质仅由它们的几何形状而不是物理性质定义），它们遵循不同的规则。通过算例说明了一般的各向同性自组织假设存在严重的缺陷。作者对表面的主要叶理的研究已经提供了一个关键结果，Rozoy将其应用于解决广义相对论中液体球对称性的一个长期存在的问题，以及将平面流体流动的流线与表面理论联系起来。

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.