Variational Problems and Nonlinear Equations in Geometry

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The interaction of geometry and analysis dates back to at least the eighteenth century, and yet continues to be an important and highly active field of mathematical research. The classical subject of geometry grew out of our desire to understand certain properties of the physical world such as angles, distances and properties of certain shapes. Differential geometry was developed to use the tools of calculus to understand the geometry of curved spaces--for example, the curvature of space by matter as predicted by general relativity, or the properties of soap bubbles (which turn out to be related to the equations describing black holes). In the same way that Descartes realized that plane geometry can be studied using algebra, so differential geometry can be studied using techniques from analysis, especially differential equations. The research in this project involves disparate problems from geometry and mathematical physics but are united by the role played by mathematical analysis in their study. The functional determinant of an elliptic operator is a problem originating in spectral theory and mathematical physics, and the analysis of the particular problem the PI considers is a variational problem leading to a fourth order elliptic equation. The associated Lagrangian is unbounded, and the existence of solutions and their qualitative properties is highly nontrivial. Similar equations are used to model the properties of thin films. Another problem the PI studies concerns the moduli space of Riemannian metrics that are critical points of functionals given by the integral of quadratic curvature quantities. In work of the PI with J. Viaclovsky, they constructed new examples of critical points, but this construction naturally leads to various conjectures about the moduli space of solutions. An important example of a quadratic curvature functional is the Weyl functional. Critical points are called Bach-flat metrics, and include important examples such as self-dual metrics. One moduli space problem the PI studies is a question of Singer about the linearized problem for self-dual manifolds of positive scalar curvature.