Heat Flow for the Yang-Mills-Higgs Field and the Hermitian Yang-Mills-Higgs Metric

For a parameter λ > 0, we study a type of vortex equations, which generalize the well-known Hermitian-Einstein equation, for a connection A and a section φ of a holomorphic vector bundle E over a Kähler manifold X. We establish a global existence of smooth solutions to heat flow for a self-dual Yang-Mills-Higgs field on E. Assuming the λ-stability of (E, φ), we prove the existence of the Hermitian Yang-Mills-Higgs metric on the holomorphic bundle E by studying the limiting behaviour of the gauge flow.