Nonlinear Problems in Geometry

摘要：项目编号：dms -0072242项目负责人：Joel spruck项目负责人提出以一种新颖的方式研究黎曼几何中的一些经典问题，这些问题可以用完全非线性椭圆方程（如蒙日-安培方程或平均曲率方程）来描述，或者与它们有很强的联系。这些问题包括经典尖锐等周不等式在负弯曲黎曼流形上的推广，在无穷远处具有规定边界的双曲空间中具有恒定平均曲率的超曲面，以及确定是否可以通过嵌入欧几里德三维空间来实现任何局部黎曼度量的问题。最后一个问题的有趣之处不仅在于其几何内容，而且在于计算机视觉中重要的具体问题。我们的研究是由具体的几何和物理问题驱动的，这些问题是纯粹和应用科学家的基本兴趣。控制我们基因中蛋白质结构的基本几何设计原理（变分原理）是什么？我们能否找到一种快速的计算算法来预测这种结构？我们怎样才能制造出更好的彩色电脑屏幕和智能相机。这些问题和许多其他问题取决于对曲面几何的深刻理解，以及描述它们如何在空间中扭曲、转动和移动的复杂非线性方程。这些问题的有效解决涉及几何、偏微分方程和高速计算的研究。

AbstractAward: DMS-0072242Principal Investigator: Joel SpruckThe principal investigator proposes to study a number ofclassical problems in Riemannian geometry that are related inthat they may be described by, or have a strong connection with,fully nonlinear elliptic equations such as Monge-Ampere equationsor mean curvature equations in some novel way. These includeextensions of the classical sharp isoperimetric inequality tonegatively curved Riemannian manifolds, hypersurfaces of constantmean curvature in hyperbolic space with prescribed boundary atinfinity, and the problem of deciding if any local Riemannianmetric can be realized by an embedding into Euclidean threedimensional space. This last problem is interesting not only forits geometric content but also for important concrete problems incomputer vision.Our research is motivated by concrete geometric and physicalproblems that are of basic interest to pure and appliedscientists. What are the basic geometric design principles(variational principles) that govern the structure of the proteinin our genes and can we find a fast computational algorithm topredict this structure. How can we make better color computerscreens and smart cameras. These questions and numerous othersdepend upon a deep understanding of the geometry of surfaces andthe complicated nonlinear equations that describe how they twistand turn and move about in space. The effective solution of theseproblems involves the study of geometry, partial differentialequations and high speed computation.