Differential Geometry and Topology of Riemannian Submanifolds

黎曼子流形的微分几何和拓扑

• 批准号: 0806305
• 负责人: Mohammad Ghomi
• 金额: $ 11.41万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0806305&HistoricalAwards=false
• 关键词: Differential; Geometry; Topology; Riemannian; Submanifolds

The PI is interested primarily in classical problems involving curves and surfaces in Euclidean space, and more generally Riemannian submanifolds. Although PI's research in this area, which includes joint work with more than a dozen collaborators, spans a wide range of topics, there are a number of recurring themes such as various notions of convexity or optmization, and the interaction between geometry and topology, which permeate throughout his work. More specifically, a typical problem is how restrictions on curvature, or various boundary conditions, influence the global shape of a curve or a hypersurface. These investigations comprise the following interelated categories: (i) Structure of locally convex hypersurfaces with boundary, including connections with Monge-Ampere equations, Alexandrov spaces with curvature bounded below, and a question of Yau; (ii) Applications of the h-principle, in the sense of Gromov, to embeddings with prescribed curvature, including knots with constant curvature or torsion; (iii) Riemannian four vertex theorems for surfaces with boundary and space curves; (iv) Capillary surfaces and generalizations of the classical isoperimetric inequality, via sharp estimates for total curvature of hypersurfaces with convex boundary;(v) Shadows on illuminated hypersurfaces and their application to geometric variational problems; (vi) Local and global isometric embedding problems; (vii) The relation between the intrinsic diameter and area of convex surfaces.Curves and surfaces are to geometry what numbers are to algebra. They form the basic ingredients of our visual perception and inspire the development of far reaching mathematical tools. Yet despite centuries of pure study, and an abundance of potential applications, there are still many fundamental open problems in this area which are strikingly intuitive and elementary to state. Moreover, technological shortcomings, such as the inability of present day computers to reliably recognize a human face, further illustrate the deficiencies in our understanding of the concept of shape. The PI believes that focusing on classical problems in submanifold geometry and topology is likely to stimulate useful developments in pure mathematics, or lead to wider applications in science and technology. For instance, those aspects of the PI's work dealing with shadows on illuminated surfaces are motivated in part by a study of soap films, and have connections to computer vision (the ``shape from shading" problems); The investigations on knots may be of interest in studying DNA; Calculating the intrinsic diameter of convex bodies is of interest in motion planing and robotics; While studying isoperimetric problems and capillary surfaces have been a significant source of enrichment in the calculus of variations. Still, the greatest impact of the proposed activity could be discovery of unexpected phenomena, or new connections between various fields.

PI主要对欧几里得空间中的曲线和曲面以及更一般的黎曼子流形的经典问题感兴趣。虽然PI在这一领域的研究，其中包括与十几个合作者的联合工作，跨越了广泛的主题，有一些反复出现的主题，如凸性或optmization的各种概念，以及几何和拓扑之间的相互作用，渗透在他的工作。更具体地说，一个典型的问题是如何限制曲率，或各种边界条件，影响曲线或超曲面的整体形状。 这些调查 包括以下相互关联的类别：（i）结构的局部凸超曲面的边界，包括与Monge-Ampere方程，Alexandrov空间的曲率有界下，和一个问题的丘;（ii）应用的h-原则，在意义上的Gromov，嵌入与规定的曲率，包括纽结与常曲率或挠;（iii）具有边界和空间曲线的曲面的黎曼四顶点定理;（iv）毛细曲面和经典等周不等式的推广，通过凸边界超曲面的全曲率的精确估计;（v）光照超曲面上的阴影及其在几何变分问题中的应用;（vi）局部和整体等距嵌入问题;（vii）凸曲面的内禀直径与面积之间的关系曲线和曲面之于几何，就像数之于代数一样。它们构成了我们视觉感知的基本成分，并激发了深远的数学工具的发展。然而，尽管有几个世纪的纯研究和丰富的潜在应用，在这个领域仍然有许多基本的开放问题，这些问题是非常直观和基本的。此外，技术上的缺陷，如目前的计算机无法可靠地识别人脸，进一步说明了我们对形状概念的理解存在缺陷。PI认为，专注于子流形几何和拓扑学中的经典问题可能会刺激纯数学的有用发展，或导致科学和技术中更广泛的应用。例如，PI处理照明表面上阴影的工作的那些方面部分是由肥皂膜的研究激发的，并且与计算机视觉有关（"shape from shading”问题）;对节的研究可能对研究DNA有意义;计算凸体的内直径在运动规划和机器人学中有意义;虽然研究等周问题和毛细管表面一直是一个重要的来源丰富 在变分法中。尽管如此，拟议活动的最大影响可能是发现意想不到的现象，或不同领域之间的新联系。