Principal Investigator: Xiaofeng Sun
In this project, the principal investigator proposed to study the geometry of moduli spaces of Riemann surfaces and stable vector bundles over a Riemann surface. These moduli spaces are the center of many branches of mathematics and physics. Continuing his previous works, the PI first proposed to prove the Mumford goodness of several canonical metrics including the Ricci metric and the Kahler-Einstein metric on the moduli space of Riemann surfaces. The goodness of these metrics will allow one to apply Chern-Weil theory to these moduli spaces which are noncompact. The PI hope to develop a method to study the intersection theory on the moduli spaces via differential geometry by using the goodness. To study the goodness of the Kahler-Einstein metric, except for the continuity method, the PI proposed to investigate the Kahler-Ricci flow on the moduli spaces. More generally, the PI proposed to study the Kahler-Ricci flow on non-compact and quasi-projective manifolds where the potential function of the initial metric is unbounded with mild growth. The PI proposed to prove that the Kahler class and the goodness of the initial metric are preserved under the Kahler-Ricci flow. Also, the long time existence and convergence of the Kahler- Ricci flow will be studied. Furthermore, the PI proposed to investigate the negativity of various metrics on this moduli space and harmonic map theory which will lead to various vanishing and rigidity theorems. The PI also proposed to extend his methods to the moduli space of stable bundles and study vanishing theorems by using canonical metrics.
The theory of moduli space of Riemann surfaces has many application in mathematics, physics and computer graphics. The intersection theory on the moduli spaces is one of the key parts in the modern string theory. Furthermore, it leads to significant improvements in image recognition, compression and decompression. The traditional method is to triangulate a surface by picking millions of vertices and record the combinatoric data. Thus a huge amount of calculations are needed. In differential geometry, the Torelli theorem tells us that the complex structure of a surface is determined by its Hodge structure. Thus one only need to record the data of the Hodge structure of a surface in order to reconstruct the surface. The size of the data of the Hodge structure is much smaller than the size of the data from triangulation. Thus a numerically efficient method to link a surface and its Hodge structure will dramatically simplify the procedure of image processing and improve its efficiency and accuracy.
