I see a statement of the theorem in Jonathan Evans' lecture note 13
An indecomposable Hermitian holomorphic vector bundle $\mathcal E$ on a Riemann surface $M$ is stable if and only if there is a compatible unitary connection on $\mathcal E$ with constant central cervature $$\star F_\nabla=-2\pi i\mu(\mathcal E)$$
Some notations: $F_\nabla$ is the curvature of the connection $\nabla$, $\mu(\mathcal E)=\frac{c_1(\mathcal E)}{\text{rank}(\mathcal E)}$ is the slope of $\mathcal E$.
Now I do not understand the statement, because from Atiyah & Bott's paper, The Yang-Mills Equations over Riemann Surfaces sec 5, we can identify the space of all unitary connection on $E$ and that of all holomorphic structures over $E$, where $E$ is the underlying complex vector bundle of $\mathcal E$. So now we are given a holomorphic structure $\mathcal E$, I think the unitary connection is uniquely determined. So is this theorem telling us that this unique-determined connection has this property?