Comparison Geometry and Rigidity

该项目的总体主题是比较几何学，这是一系列技术的集合，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、质量和其他几何测量。 当其中一个空间是一个很好理解的“模型”空间时，比较几何通常提供对其他数学领域（例如，在庞加莱猜想的解决中发挥重要作用）以及其他科学有用的最佳估计。 在这个项目中，主要研究者将使用比较几何来估计光滑流形中域的基本间隙。 这个间隙是拉普拉斯算子的前两个本征值之间的差异：为了解释这一点，拉普拉斯算子经常出现在数学物理中，当模拟波通过各种介质的传播时，本征值是域的谐振频率。 质量间隙也与量子力学有关，它代表了将粒子从基态激发到下一个能级所需的能量。因此，这个项目的进展将在数学和物理学方面产生潜在的好处。 PI还计划估计非光滑空间（称为度量测度空间）的体积熵，PI关于熵刚性的结果将与最佳传输、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，支持研究生和青年研究人员的教育和培训。这个项目是关于几个问题围绕比较几何。通过与一个好的模型进行比较，比较几何通常提供最佳估计。 例如，佩雷尔曼的约化体积单调性，这是他在庞加莱和几何化结构方面工作的基本工具，可以被视为Bishop-Gromov体积比较到Ricci流的推广。Andrews和Clutterbuck通过证明第一本征函数比模型更对数凹来证明基本间隙猜想。 PI将研究局部对称空间中凸域上拉普拉斯算子（具有Dirichlet边界条件）的特征值和基本间隙比较估计，以及具有积分Ricci曲率下限和非光滑扩展的闭流形（具有Neumann边界条件）。 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）以及具有曲率下界的度量测度空间的体积熵比较和刚性。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）中起着至关重要的作用，研究非光滑情况将应用于光滑流形的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价值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.