# The Yang-Mills flow for connections

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I consider a riemannian manifold $(M,g)$ with a fiber bundle $E$ equipped by one parameter family of connections $\nabla^t$. The curvature of $\nabla^t$ is $R(\nabla^t)$ and the codifferential is $\delta_{\nabla^t} = * \circ d_{\nabla^t} \circ *$, with $*$ the Hodge operator and $d_{\nabla^t}$ the differential of the connection on the forms $\Lambda^*(M) \bigotimes End(E)$. The Yang-Mills flow is then defined by the formula:

$$\frac {\partial }{\partial t} (\nabla^t ) = \delta_{\nabla^t} (R (\nabla^t))$$

I have in each term of the equality a 1-form with values in the endomorphisms of $E$. The fixed points of the flow are the connections with harmonic curvature (as the curvature is closed). They are the instantons in physics for $SU(2)$-connections over the space-time manifold.

Is such a flow well-defined for a short interval of the time near the initial connection? When is the flow convergent?

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