Geometric Inequalities and Fully Nonlinear Elliptic Equations

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The Principal Investigator (PI) will study various geometric inequalities and their corresponding nonlinear partial differential equations in the context of conformal geometry and the geometry of submanifolds. One theme plans to investigate the effect of the higher order curvatures on the validity of the isoperimetric inequality. In particular, she intends to quantitatively analyze the interaction between the Q-curvature and the isoperimetric constant. The PI also proposes to study curvature inequalities of different orders for embedded submanifolds. These inequalities, originally considered in the context of convex geometry, have recently known to be valid on a very large class of non-convex domains. The PI's investigation strives to look for the full generality of such inequalities. In the meanwhile, she also aims to develop new skills to understand the corresponding fully nonlinear elliptic partial differential equations that arise naturally in the problem.In the proposed study, the PI's research interest lies at the intersection of conformal geometry, the geometry of submanifolds and partial differential equations (PDEs). The study of geometric inequalities and geometric PDEs focuses on conformal invariants, which form an important machinery from physicists' point of view and have found applications to fundamental principles in mathematical physics. The research project to generalize classical results of convex geometry will improve our understanding on the rigidity of the established theory and will shed light on a greatly larger scope of its application.