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

TL;DR

This paper investigates the blow-up behavior of solutions to singular Liouville equations with collapsing poles, establishing quantization and precise estimates that relate to geometric applications in hyperbolic surface immersions.

Contribution

It provides a detailed analysis of blow-up phenomena at collapsing poles, extending quantization results and deriving point-wise estimates for solutions.

Findings

Quantization property of blow-up mass is preserved at collapsing poles.

Precise point-wise estimates are obtained for minimal blow-up mass cases.

Results connect blow-up analysis to geometric problems involving surface immersions.

Abstract

We analyse 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 point-wise estimates when blow-up occurs with the least blow-up mass. Interestingly, such estimates express the exact analogue of those obtained for "bubble" solutions of "regular" Liouville equations, when the "collapsing" Dirac measures are neglected. Such information will be used to describe the asymptotic behaviour of minimizers of the Donaldson functional introduced by Goncalves and Uhlenbeck (2007), yielding to mean curvature 1-immersions of surfaces into hyperbolic 3-manifolds.