A mean field type equation on vector bundles

The mean field type equation is a basic problem of mathematics. Up to now, far too little attention has been paid to the mean field type equation on vector bundles. Recently, some existence results for a mean field type equation on line bundles over a closed Riemann surface were shown in our prior work. In the present paper, we generalize the previous results to vector bundles. Precisely, we prove the existence of minimizing solutions to a mean field type equation on vector bundles in the subcritical case by a direct method of calculus of variations and obtain a sufficient condition for the existence result in the critical case via the method of blow-up analysis. Taking account of the additional terms corresponding to the distributional decomposition of the bundle Laplace–Beltrami operator, we need delicate blow-up analysis and elaborate estimates.