Hermitian-Yang-Mills Metrics on Hilbert Bundles and in the Space of Kahler Potentials
Abstract
The two main results in this thesis have a common point: Hermitian- Yang- Mills (HYM) metrics. In the first result, we address a Dirichlet problem for the HYM equations in bundles of infinite rank over Riemann surfaces. The solvability has been known since the work of Donaldson [Don92] and Coifman-Semmes [CS93], but only for bundles of finite rank. So the novelty of our first result is to show how to deal with infinite rank bundles. The key is an a priori estimate obtained from special feature of the HYM equation. In the second result, we take on the topic of the so-called “geometric quantization.” This is a vast subject. In one of its instances the aim is to approximate the space of Kahler potentials by a sequence of finite dimensional spaces. The approximation of a point or a geodesic in the space of Kahler potentials is well-known, and it has many applications in Kahler geometry. Our second result concerns the approximation of a Wess-Zumino-Witten type equation in the space of Kahler potentials via HYM equations, and it is an extension of the point/geodesic approximation.
Degree
Ph.D.
Advisors
Lempert, Purdue University.
Subject Area
Mathematics
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