A minimal mass blow-up solution on a nonlinear quantum star graph

We construct a finite-time blow-up solution to the mass-critical focusing nonlinear Schr\"odinger equation on a metric star graph with an arbitrary number of edges. We show that all solutions are global if their mass is smaller than an explicit constant, called "minimal mass". We then construct a solution with minimal mass and arbitrary energy, which blows up in finite time at the vertex of the star graph. The blow-up profile and blow-up speed are explicitly characterized. The main novelty of the paper is the construction of the blow-up profile in time-dependent domains of singularly perturbed Laplacians.