Differential Geometry and Topology of Riemannian Submanifolds

PI主要研究欧氏空间中涉及曲线和曲面的经典问题，以及更广泛的黎曼子流形。尽管派在这一领域的研究，包括与十几位合作者的联合工作，跨越了广泛的主题，但有许多反复出现的主题，如各种凸性或最优化的概念，以及几何和拓扑之间的相互作用，这些贯穿于他的作品中。更具体地说，一个典型的问题是曲率限制或各种边界条件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范畴：(I)带边界的局部凸超曲面的结构，包括与Monge-Ampere方程的联系，曲率有界的Alexandrov空间，以及Yau问题；(Ii)h-原理在Gromov意义下对具有指定曲率的嵌入的应用，包括具有常曲率或扭转的纽结；(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全曲率的精确估计；(V)照明超曲面上的阴影及其在几何变分问题中的应用；(Vi)局部和整体等距嵌入问题；(Vii)凸面的内径和面积之间的关系。曲线和曲面之于几何，就像数字之于代数。它们构成了我们视觉感知的基本成分，并启发了影响深远的数学工具的发展。然而，尽管经过了几个世纪的纯粹研究和丰富的潜在应用，这一领域仍然有许多基本的开放问题，这些问题对于国家来说是惊人的直观和基本的。此外，技术上的缺陷，例如现在的计算机不能可靠地识别人脸，进一步说明了我们对形状概念的理解的不足。国际数学联合会认为，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，或者导致更广泛的科学和技术应用。例如，PI处理照亮表面上阴影的工作的那些方面部分是由肥皂膜的研究推动的，并与计算机视觉有关(从阴影到形状的问题)；对结点的研究可能对研究DNA感兴趣；计算凸体的内部直径是运动规划和机器人学的兴趣；而研究等周问题和毛细表面一直是变分演算丰富的重要来源。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。

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.