Metric geometry and analysis on Einstein manifolds

Metric Riemannian geometry is a central subject in modern mathematics. The original concept dates back to Bernhard Riemann's famous Habilitation lecture "Ueber die hypothesen, welche der Geometrie zu Grunde liegen" (On the hypotheses which lie at the bases of geometry) delivered on 10 June 1854. The revolutionary creations in this lecture profoundly changed the global landscape of geometry. Specifically, Riemann proposed a novel strategy to generalize the geometry of surfaces to higher dimensions which he called Mannigfaltigkeiten (manifolds). A large variety of new notions and concepts were created: these include the notion of curvature which quantitatively measures how a space is curved, and the notion of geodesic which is a length-minimizing path connecting two points on a manifold. The studies of the metric structures of manifolds, what we now call metric Riemannian geometry, primarily focuses on the interplay between the global geometry of the underlying space and the metric structure, namely how the distance between two points can be realized or measured. This project is mainly concerned with the metric geometry of Einstein manifolds where the metric structures satisfy the Einstein equation in the theory of general relativity. The PI will integrate their research with training and mentorship at a variety of levels. This includes organizing summer workshops and mathematical retreats on Riemannian geometry; complex geometry and theoretical physics, and designing and developing new research oriented courses for undergraduate students.

This project investigates the degenerations and quantitative behaviors of Einstein manifolds. In joint work with Song Sun, the PI has been working on the collapsing geometry of Einstein manifolds with special holonomy, leading to two major breakthroughs in the field: a complete classification of the Gromov-Hausdorff limits of the Einstein metrics on the K3 manifold, and a complete classification of asymptotic model geometries of gravitational instantons. The latter can be regarded as the bubble limits of the degenerating Einstein metrics on the K3 manifold. Building on this background, the PI will proceed to analogous questions in higher dimensions and investigate geometric structures for the degenerating Einstein metrics with generic holonomy in that setting. With a group of collaborators, the PI will also make advances in more refined geometry and moduli space problems regarding complete Calabi-Yau metrics. In a third direction, the PI will investigate the geometry and analysis of Poincare-Einstein spaces, which originated from the AdS/CFT correspondence in mathematical physics. The PI will focus on the singularity behaviors of degenerate operators on Poincare-Einstein manifolds, geometric finiteness and quantitative rigidity of Poincare-Einstein metrics, as well as regularity and degeneration theory of Poincare-Einstein metrics.

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.

Award Number: 2550348
Principal Investigator: Ruobing Zhang
Funds Obligated: $74,838
State: CA