Time-dependent Hermitian Yang–Mills flow on surfaces

Abstract

We investigate a generalisation of Hermitian Yang–Mills flow in which the base metric itself is allowed to depend on the time-parameter. For technical reasons we restrict our attention to the case of a holomorphic vector bundles E over compact Riemann surfaces which is assumed to be slope stable with respect to a fixed Kahler class \(\tau \) on the base. We show that if \(\omega (t)\) is a family of Kahler metrics in \(\tau \) converging to a limit metric \(\omega _\infty \) at an exponential rate, then starting at a smooth initial Hermitian metric \(h_0\) on E, the Hermitian Yang–Mills flow with respect to the time-dependent metric \(\omega (t)\) admits a unique long-time solution h(t) converging exponentially to a \(\omega _\infty \)-Hermite–Einstein metric. The proof utilises an extension of Donadson’s techniques used to treat the case where the base metrics does not depend on the time-parameter.