Canonical Kahler metrics and complex Monge-Ampere equations

The project will focus on addressing open problems in geometric analysis and exploring their applications in various fields such as geometry, topology and mathematical physics. These problems play a central role in active areas of research in mathematics, including differential geometry, partial differential equations (PDE), and high-dimensional supergravity. Given the interdisciplinary nature of this project, it will foster collaborations among researchers from various disciplines, and the outcomes of the project will introduce novel approaches and provide valuable insights into the analytic study of the geometry of singular varieties. An important objective of the project is to establish a foundation for the integration of research and education, enriching the mathematics curriculum and enhancing the mathematics education at Rutgers - Newark. In line with this objective, the Principal Investigator (PI) will organize seminars and deliver lectures, aiming to contribute towards the advancement of mathematics education nationwide. The PI will also engage in mentoring at at high school, undergraduate, and graduate levels. The PI will continue to develop novel approaches in the regularity theory for linear and fully nonlinear PDEs on complex manifolds, with a specific focus on the complex Monge-Ampere equations and the associated Kahler metrics. The geometry of these metrics will be investigated from both analytic and geometric perspectives. An emphasis will be placed on studying the degeneration of a family of Kahler metrics, including the geometric convergence of Kahler-Ricci flow and other flows arising from geometry and physics. To this end, the PI will advance the techniques of auxiliary differential equations, aiming to analyze the compactness of the space of the family of Kahler metrics. Along this path, it is expected that new analytic tools such as uniform Poincare and Sobolev inequalities, as well as heat kernel estimates, will be developed. Furthermore, combined with techniques from complex geometry and algebraic geometry, these tools will be employed to investigate the asymptotic behavior of metrics near singularities. In addition, the PI will continue to explore the parabolic approach, introduced by the PI and collaborators, in high-dimensional supergravity. This exploration aims to discover new ansatz and construct new solutions to the coupled systems, thereby deepening the understanding of the underlying space. 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.