Executive Summary : | This projects consists of three main parts: In the first part, the PI proposes a program to prove existence and uniqueness of weak solutions to a large class of non-linear partial differential equations on compact Kahler manifolds which includes equations such as the J-equation and the deformed Hermitian-Yang-Mills equations. It is further proposed that the PI will investigate the convergence of certain natural parabolic flows to these weak solutions, and also study the blow-up profiles of such flows. The proposed program seeks to further our understanding of the relation between lack of positivity in algebraic geometry and singularities of solutions to non-linear PDEs. In the second part, the PI proposes to investigate whether a compact Kahler manifold with bi-sectional curvature lower bounded by one, and almost maximal diameter is Gromov-Hausdorff close to the complex projective space endowed with the Fubini-Study metric? Such a result would be of great interest to geometers considering the analogous statement for general Riemannian manifolds with lower sectional curvature bounds is known to be false. In the third part of the project the PI proposes to study the completeness of some Kahler-Einstein metrics on complements of isolated log-canonical singularities, and proposes a classification of these singularities (at least on surfaces) via an analysis of the tangent cones at infinity. This will further our understanding of the geometry of collapsing Kahler-Einstein metrics of negative scalar curvature, and should be thought of as a higher dimensional analog of the degeneration of hyperbolic metrics as a Riemann surface of high genus degenerates to a nodal curve at the boundary of the Deligne-Mumford compatification. |