Problems in surface theory/ membranes/hypersurface singularities

摘要奖：DMS-0071729首席研究员：布莱恩·史密斯这项建议主要涉及表面理论的定性方面，以及它在构建一般弹性膜的完整平衡理论方面的应用。表面理论的工作正在揭示空间中光滑表面上孤立奇点(脐状)附近的主叶的先前未被怀疑的定性行为，这显著地将它们与一般的叶区分开来。这一行为解释了Caratheodory猜想，并在许多情况下证明了该猜想是一个枚举式结果。关于弹性膜的工作建立了最一般的拉伸弹性膜在外加力场作用下的平衡方程-仅适用于各向同性膜的经典Young-Laplace方程的进一步推广。这项工作的背景是膜表面的单位圆丛，它配备了膜表面上诱导度规的正则升力(Sasaki度规)；关键思想是在这个丛上构造一个矢量值1-形式，它完整地编码了膜内设置的表面张力的响应力。表面是在物理、化学和生物学中理解物质结构的基本数学实体。两种不同类型物质交界处的表面通常用弹性膜来模拟。处于平衡状态的弹性膜的任何人都可以看到膜的形状，了解所施加的力，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察量与一般膜的内表面张力之间的精确关系。由此得出的结论是，响应从来不是完全可以从可观察到的(膜的几何形状、作用力、平均表面张力)来预测的，但是某些核心行为是确定的-除了一类非常了不起的膜(其特殊性质完全由它们的几何形状而不是它们的物理性质定义)，它们遵循着一套不同的规则。算例表明，常见的各向同性假设是有严重缺陷的。作者在曲面主叶上的工作已经提供了一个关键结果，Rozoy已经应用该结果解决了广义相对论中液球对称性的一个长期存在的问题，并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.