Final Report Summary - DIFFERENTIALGEOMETR (Geometric analysis, complex geometry and gauge theory)
This project had two components, involving work in different, but related areas of differential geometry. One area was complex differential geometry, and specifically the study of Riemannian metrics on complex algebraic manifolds . These are somewhat like the metric tensor which is the fundamental notion in General Relativity, and there is a version of the Einstein condition, which amounts to a partial differential equation for the metric. A central problem in the field, which has been studied for 60 years or more, is to decide when this Einstein condition can be satisfied. The most difficult case is the class of “Fano” manifolds. The answer in this case can depend on the detailed form of the algebraic equations defining the manifold. The main goal of our work was to give a precise and completely general algebro-geometric criterion for the solubility of the Einstein problem, involving a notion of “K-stability”. This goal was achieved at the end of 2012, in joint work of the PI, S. Sun (research assistant funded by the project) and X. Chen. This built on a number of important technical advances, in particular in relating the theory of convergence in Riemannian geometry to complex algebraic geometry. Another research assistant, M. de Borbon, wrote a PhD thesis on a topic in this area.
The second component of the project involved the study of 7 and 8 dimensional Riemannian manifolds with exceptional holonomy groups. These also satisfy a version of the Einstein condition and there are important connections with “M-theory” in theoretical physics. The staff involved were the PI, J. Nordstrom and T. Walpuski (who completed a PhD in the area). Nordstrom’s work focused on the refinement of the “twisted connected sum” construction and on existence and singularity formation for associative submanifolds. Walpuski’s work focused on a version of the Yang-Mills equation in this context, and the failure of compactness due to “bubbling” on an associative submanifold. These are relatively undeveloped fields and the output of the project constitutes a major advance in our understanding of these special geometries.
The second component of the project involved the study of 7 and 8 dimensional Riemannian manifolds with exceptional holonomy groups. These also satisfy a version of the Einstein condition and there are important connections with “M-theory” in theoretical physics. The staff involved were the PI, J. Nordstrom and T. Walpuski (who completed a PhD in the area). Nordstrom’s work focused on the refinement of the “twisted connected sum” construction and on existence and singularity formation for associative submanifolds. Walpuski’s work focused on a version of the Yang-Mills equation in this context, and the failure of compactness due to “bubbling” on an associative submanifold. These are relatively undeveloped fields and the output of the project constitutes a major advance in our understanding of these special geometries.