Let E be a holomorphic bundle over algebra surface X, let $H$ be a Hermitian metric of $E$, recall the Hermitian-Yang-mills equation is $\wedge F_H=\lambda.1$.

Let $H_t$ be Hermitian metrics over $E$ parametrized by $t$, Donaldson in [1] consider the following flow equation: \begin{equation} H_t^{-1}\frac{\partial H_t}{\partial t}=-2i(\wedge F_{H_t}-\lambda.1),\;\;H_t|_{t=0}=H_0, \label{flow} \end{equation} for some initial metric $H_0$.

In [1], page 15, there is a note: If $E$ is indecomposable and has a solution $K$ to the Hermitian-Yang-mills equation, then for any initial condition $H_0$, the corresponding solution $H_t$ of the flow equation converges in $\mathcal{C}^{\infty}$ to $K$ as $t\to\infty$. In addition, consider the distance function $\sigma$ between two metric $H_t,K$, $\sigma(H_t,K):=Tr(H_t^{-1}K)+Tr(K^{-1}H_t)-2\;\mathrm{rank}\;E$, then we have a bound \begin{equation} \|(\frac{\partial}{\partial t}+\Delta)\sigma(K,H_t)\|_{L^1}\leq -const.\|\sigma(K,H_t)\|_{L^1} \end{equation} and $\sigma$ decays exponentially.

My question is how to verify these two claims:

(A)If a solution $K$ exists, then the flow convergence to the solution in $\mathcal{C}^{\infty}$.

(B)This convergence is exponentially decays.

[1] S.Donaldson, Anti Self-Dual Yang-Mills Connections over Complex Algebraic Surfaces and Stable Vector Bundles


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.