Comparison Geometry and Rigidity

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

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.