CANONICAL METRICS ON KAHLER AND RIEMANNIAN MANIFOLDS AND THEIR MODULI

Project Details

Description

It is a fundamental problem to construct appropriate spaces of various geometric structures in both algebraic and differential geometry. These spaces, referred to as moduli spaces, have played a fundamental role in mathematical subjects ranging from analysis, geometry, and topology to number theory. More than a century ago, Riemann, Poincare, and Hilbert pioneered the study of 'canonical' metric structures on a surface. Through the connection to the Einstein equations, the projects under investigation will help understand astronomy and Cosmology. The methods of analyzing Einstein type equations in this project will also lead to applications in Engineering and economics. The principal investigator organizes and participates in the integrated research/education programs that aim to attract students from under-represented groups to the study of more advanced mathematics subjects such as geometric analysis, thus helping to improve the overall education level of the nation. The PI will investigate the following projects, all of which emerge from the study of the algebraic/analytic aspects of the moduli spaces. First, he will study the geometry of the moduli space of polarized Kahler manifolds via algebraic and analytic means (e.g., the positivity of the line bundles and heights over the moduli space). Second, he will study the effective scope of geometric invariant theory (GIT) invented by Hilbert and Mumford on the construction of algebraic moduli based on the work of the PI and his collaborators. In particular, the PI investigates more examples in order to develop a general framework that is more flexible than the classical GIT. Third, he will study the limiting behavior of Kahler-Ricci flow on low-dimensional manifolds, in particular the situation when the complex structure jumps. Fourth, he will study the canonical embedding of Riemannian manifolds into the infinite-dimensional unit sphere and relate the extrinsic geometry of the embedding to the canonical metrics on the underlying Riemannian manifold. This is intended as the first step in attempting to unify the similarities between conformal and Kahler geometry. The projects proposed above will apply both analytic and algebraic tools coming from algebraic/differential geometry as well as the new ideas introduced by the principal investigator and his collaborators over the years.
StatusFinished
Effective start/end date9/1/168/31/19

Funding

  • National Science Foundation (National Science Foundation (NSF))

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