Conformal invariants, Einstein manifolds, and their minimal submanifolds

Einstein manifolds and minimal submanifolds, which describe the vacuum spacetimes in Einstein's theory of general relativity and soap films, respectively, are objects of fundamental interest in mathematics and physics. This project will use methods from conformal geometry, where one is allowed to change lengths but not angles, to construct and classify geometric invariants which can control the possible behaviors of Einstein manifolds and their minimal submanifolds. Particular emphasis will be placed on conformally compact Einstein manifolds and their minimal submanifolds, which are fundamental in string theory. This project includes research problems designed for undergraduate and graduate students that will help quickly integrate them into high-level mathematics research. This project is split into three subprojects. The first subproject will develop the PI's recently developed notion of straightening into a classification of global conformal invariants of Einstein manifolds and of their minimal submanifolds. The second subproject uses these invariants to prove gap theorems that isolate spaceforms and their totally geodesic submanifolds via higher-order analogues of the L^2-norm of the Weyl tensor and the Willmore energy of a surface. It will also use these invariants to construct invariants whose positivity obstructs the existence of an Einstein metric on a given smooth manifold and develop tools to compute those invariants. The third subproject will develop new analytic tools for studying Q-curvature, including a strong maximum principle for higher-order GJMS operators and an Obata-type classification of conformally Einstein metrics with constant Q-curvature. 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.