Nonlinear PDEs in Complex Geometry and Physics

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This proposed research aims to further develop our knowledge of differential geometry and nonlinear partial differential equations. Our approach is to use techniques from the analysis of nonlinear partial differential equations to study Riemannian metric tensors on manifolds. Optimal Riemannian metrics generally satisfy a constraint equation on their curvature tensor which can be expressed as a nonlinear PDE. I am particularly interested in equations coming from mathematical physics and complex geometry, in both Kahler and non-Kahler settings. The principle that Riemannian metrics can be used to describe the underlying space is found throughout mathematics, beginning with the uniformization theorem of complex analysis, and since appearing in the theory of Hermitian-Yang-Mills connections on stable vector bundles, the Kodaira embedding theorem, the Poincaré conjecture, and the Yau-Tian-Donaldson conjecture, just to name a few. This proposal continues this tradition while bringing in new equations introduced in theoretical physics. Beyond physical applications, the study of metrics subject to a curvature constraint links geometry and the field of partial differential equations. On one hand, differential geometry provides interesting examples of nonlinear equations with deep structural properties, and on the other hand, the development of new techniques in elliptic and parabolic differential equations often leads to breakthroughs in differential geometry. The first project concerns the construction of new solutions to the Hull-Strominger system. This is a system of differential equations proposed by theoretical physicists as a model for the heterotic string; furthermore, the Hermitian metrics involved may have nonzero torsion, which makes them interesting from the point of view of non-Kahler complex geometry. The second project concerns the analysis of the Anomaly flow. This geometric flow can be viewed as an analog of the Ricci flow adapted to the geometric setting of Calabi-Yau manifolds with torsion. The third project concerns the pure PDE problem of obtaining a priori estimates on solutions of certain fully nonlinear elliptic equations. These equations are inspired by geometry, and examples include the complex Monge-Ampere equation and the k-th Hessian equation. In summary, the goal of this research is to advance the analysis of nonlinear equations in differential geometry, with a main focus on equations from theoretical physics.