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On a compact Riemann surface with a metric, there exists a Green’s function $C ln(d(x,y)^2)\leq G(x,y)\leq 0$ satisfying $u=\int u+ \int G(x,y) u(y) dy$.

Suppose $(E,h)$ is a Hermitian holomorphic bundle on a compact Riemann surface, is there a negative-definite Green’s function $G(x,y)$ satisfying a similar representation formula? (It probably exists, but my worry is about an upper bound on it.) Is there a reference for the same?

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  • $\begingroup$ Given that $G$ is a section in $E \boxtimes E^*$, what does it mean for it to be negative-definite? $\endgroup$
    – Alex M.
    Commented Dec 1, 2023 at 15:48

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