Partial differential equations in conformal and CR geometry

点击翻译按钮获取中文摘要

This project will study the higher order Q-curvature equations in conformal geometry and CR geometry. These equations are closely related to the Gauss-Bonnet integral in the case of conformal geometry and to the logarithm term in the Szego kernel in the case of CR geometry. The previous work of the principal investigators has established the significance of these equations in helping to understand the geometry of the spaces in questions. The current project takes further steps to explore the analytic development of these Q-curvature equations and basic questions relating to the positivity of the underlying operators. In the case of CR geometry, the research seeks to relate the Q-curvature equation to the notion of mass for a CR structure. Again the positivity of the underlying operator is the focus of the investigations. A successful outcome would lead to a strong characterization of the standard sphere in complex space.The principal investigators of this project have pioneered the use of a natural partial differential equation involving a larger number of differentiations as analytic tools to study the geometry of four-dimensional spaces. The scientific part of the proposal is a natural continuation of this development, where the focus is on verifying certain conjectural pictures of the space-time universe created by physicists. A good portion of the project will involve the training of the next generation of mathematicians as researchers and teachers of mathematics. The principal investigators will also continue their past record of mentoring a number of successful women mathematicians.