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 主要对涉及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。尽管 PI 在该领域的研究（包括与十几位合作者的联合工作）涵盖了广泛的主题，但仍有许多反复出现的主题，例如各种凸性或优化的概念，以及几何和拓扑之间的相互作用，渗透到他的整个作品中。更具体地说，一个典型的问题是曲率或各种边界条件的限制如何影响曲线或超曲面的全局形状。 这些研究包括以下相互关联的类别：（i）具有边界的局部凸超曲面的结构，包括与 Monge-Ampere 方程、曲率下界的 Alexandrov 空间以及 Yau 问题的联系； (ii) 格罗莫夫意义上的 h 原理在具有规定曲率的嵌入中的应用，包括具有恒定曲率或扭转的结； (iii) 具有边界和空间曲线的曲面的黎曼四顶点定理； (iv) 毛细管曲面和经典等周不等式的推广，通过对具有凸边界的超曲面的总曲率进行锐估计；(v) 照明超曲面上的阴影及其在几何变分问题中的应用； (vi) 局部和全局等距嵌入问题； (vii) 凸曲面的内径和面积之间的关系。曲线和曲面对于几何学就像数字对于代数一样。它们构成了我们视觉感知的基本成分，并激发了影响深远的数学工具的发展。然而，尽管经过几个世纪的纯粹研究和大量的潜在应用，该领域仍然存在许多基本的开放问题，这些问题非常直观且易于陈述。此外，技术缺陷，例如当今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理解的缺陷。 PI 认为，关注子流形几何和拓扑中的经典问题可能会刺激纯数学的有益发展，或导致科学和技术中更广泛的应用。例如，PI 处理照明表面阴影的工作部分是由对肥皂膜的研究推动的，并且与计算机视觉（“阴影形状”问题）有联系；对结的研究可能是对研究 DNA 感兴趣；计算凸体的内在直径对运动规划和机器人学感兴趣；而研究等周问题和毛细管表面一直是丰富变异计算的重要来源。拟议的活动可能是发现意想不到的现象，或者不同领域之间的新联系。