The Heat Equation Approach to Hermitian-Einstein Metrics on Stable Bundles

Abstract

In this chapter we discuss here the problem of the existence of Hermitian-Einstein metrics on stable bundles. The case of stable bundles over compact complex curves was done by Narasimhan and Seshadri [N-S] and alternative proof later given by Donaldson [D1]. Donaldson [D2] treated the case of stable bundles over compact algebraic surfaces by using the heat equation method. Recently a paper of Uhlenbeck and Yau [U-Y] dealt with the problem for stable bundles over a compact Kahler manifold by using the continuity method. A recent preprint of N. P. Buchdahl [Bu] considered the case of stable bundles over compact complex surfaces that admit a ∂†̄-closed positive definite (1.1)-form. A most recent preprint of Donaldson [D3] carried over his treatment of the surface case to higher dimension. Simpson [Si] in his Harvard University dissertation under the direction of W. Schmid proved the existence of Hermitian-Einstein metrics for stable systems of Hodge bundles without the restriction of algebraicity.