Nonlinear Problems in Geometry

摘要奖：DMS-0072242主要研究人员：Joel Spruck主要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的椭圆型方程来描述，或以某种新颖的方式与完全非线性的椭圆型方程如Monge-Ampere方程或平均曲率方程相关联。其中包括经典尖锐等周不等式的推广，负弯曲黎曼流形的推广，具有给定边界无穷远的双曲空间中常平均曲率的超曲面，以及判定局部黎曼度量是否可以通过嵌入到欧氏三维空间来实现的问题。最后一个问题很有趣，不仅因为它的几何内容，而且还因为计算机视觉中的重要具体问题。我们的研究是由纯粹的和应用型科学家基本感兴趣的具体几何和物理问题推动的。控制我们基因中蛋白质结构的基本几何设计原则(变分原则)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。我们如何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入理解，以及描述曲面如何在空间中扭曲、旋转和移动的复杂的非线性方程。这些问题的有效解决涉及到几何研究、偏微分方程组和高速计算。

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.