Comparison Geometry and Rigidity

这个项目的总体主题是比较几何学，这是一种技术的集合，通过比较两个不同空间的曲率，可以比较两个空间中物体的长度、面积、体积、质量和其他几何尺寸。当其中一个空间是众所周知的“模型”空间时，比较几何往往提供对其他数学领域有用的最佳估计(例如，在解决庞加莱猜想方面发挥重要作用)以及其他科学。在这个项目中，主要的研究人员将使用比较几何来估计光滑流形中区域的基本间隙。这个间隙是前两个拉普拉斯本征值之间的差异：为了解释这一点，在数学物理中，当模拟波在各种介质中的传播时，拉普拉斯经常出现，本征值是区域的共振频率。质量间隙也与量子力学有关，在量子力学中，它表示将粒子从基态激发到下一个能级所需的能量。因此，这一项目的进展将在数学和物理方面带来潜在的好处。PI还计划估计被称为度量空间的非光滑空间的体积熵，PI关于熵刚性的结果将与最优传输、信息几何和离散几何相关。该项目还将通过博士后指导和教学，支持研究生和年轻研究人员的教育和培训。这个项目涉及几个以比较几何为中心的问题。通过与一个好的模型进行比较，比较几何通常提供最优估计。例如，Perelman的约化体积单调性是他研究Poincare猜想和几何化猜想的基本工具，它可以被视为毕晓普-格罗莫夫体积与Ricci流的比较的推广。Andrews和Clutterbuck通过证明第一特征函数比模型更对数凹，证明了基本间隙猜想。PI将研究局部对称空间中凸域上的拉普拉斯(具有Dirichlet边界条件)以及具有积分Ricci曲率下界和非光滑扩张的闭流形(具有Neumann边界条件)的特征值和基本间隙比较估计。PI还将研究比较几何和Ricci流之间的关系(包括Ricci流的几乎刚性结果)，以及曲率下界度量空间的体熵比较和刚性。体积熵是紧致光滑流形的一个基本几何不变量，这个概念与动力系统中的其他熵概念密切相关。它在微分几何和几何群论(以及其他)中扮演着重要的角色，研究非光滑情况将适用于光滑流形的Gromov-Hausdorff极限。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The overall subject of this project is comparison geometry, a collection of techniques whereby comparisons of the curvature of two different spaces are related to comparisons of length, area, volume, mass and other geometric measurements of objects in the two spaces. When one of the spaces is a well-understood `model' space, comparison geometry often provides optimal estimates that are useful for other areas of mathematics (for example, playing an essential role in the resolution of the Poincare conjecture) as well as other sciences. In this project, the principal investigator will use comparison geometry to estimate the fundamental gap for domains in smooth manifolds. This gap is the difference between the first two eigenvalues of the Laplacian: to explain this, the Laplacian frequently occurs in mathematical physics when modelling the propagation of waves through various media, and the eigenvalues are the resonant frequencies of the domain. The mass gap is also related to quantum mechanics, where it represents the energy needed to excite a particle from its ground state to the next energy level. Advancement in this project will thus have potential benefits in mathematics and physics. The PI also plans to estimate the volume entropy for non-smooth spaces known as metric measure spaces, and the PI's results on entropy rigidity will be relevant to optimal transport, information geometry and discrete geometry. This project will also support education and training of graduate students and young researchers, through postdoctoral mentoring and teaching. This project is concerned with several problems centered around comparison geometry. By comparing to a good model, comparison geometry often provides optimal estimates. For example, Perelman's reduced volume monotonicity, which is the fundamental tool in his work on the Poincare and geometrization conjectures, can be viewed as generalization of Bishop-Gromov volume comparison to Ricci flow. Andrews and Clutterbuck proved the fundamental gap conjecture by showing the first eigenfunction is more log-concave than the model. The PI will study the eigenvalue and fundamental gap comparison estimates for the Laplacian (with Dirichlet boundary conditions) on a convex domain in locally symmetric spaces, and on closed manifolds (with Neumann boundary conditions) with integral Ricci curvature lower bound and non-smooth extensions. The PI will also study the relation between comparison geometry and Ricci flow (including almost rigidity results for Ricci flow) as well as volume entropy comparison and rigidity for metric measure spaces with curvature lower bounds. Volume entropy is a fundamental geometric invariant for compact smooth manifolds, and this concept is closely related to other notions of entropy found in dynamical systems. It plays an essential role in differential geometry and geometric group theory (among others), and studying the non-smooth case will have applications to Gromov-Hausdorff limits of smooth manifolds.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.