Local Uniqueness and Non-degeneracy of Blow Up Solutions To A Chern-Simons System

In this paper, we study blowup solutions of an important class of Chern-Simons systems.

We first show that when blowup of mean-field type occurs, the corresponding blowup solution is unique under natural geometric assumptions.

We also establish the non-degeneracy of the linearized system around these blowup solutions.

To prove these main results, we carry out a precise blowup analysis, so that the asymptotic description of the solutions reveals the curvature information needed for the uniqueness and non-degeneracy results.

Compared with related work on similar problems, our estimates are more delicate and technically involved.