On the blow-up analysis at collapsing poles for solutions of singular Liouville-type equations

Abstract

We analyze a blow-up sequence of solutions for Liouville-type equations involving Dirac measures with “collapsing” poles. We consider the case where blow-up occurs exactly at a point where the poles coalesce. After proving that a” quantization” property still holds for the” blow-up mass,” we obtain precise pointwise estimates when blow-up occurs with the least blow-up mass. Interestingly, such estimates express the exact analogue of those previously obtained for solutions of “regular” Liouville equations where the “collapsing” Dirac measures are neglected. Such information will be used in a forthcoming paper to describe the asymptotic behavior of minimizers of the Donaldson functional introduced by Goncalves and Uhlenbeck in 2007, yielding to mean curvature 1-immersions of surfaces into hyperbolic 3-manifolds.