In this article, the authors developed a new approach to construct static metric extensions as they arise in R. Bartnik’s conjecture [Commun. Pure Appl. Math. 39, 661–693 (1986; Zbl 0598.53045)]. They suggested a geometric flow approach, coupled to the Weyl-Papapetrou formalism for axisymmetric static solutions to the Einstein vacuum equations. The elliptic Weyl-Papapetrou system becomes a free boundary value problem in this approach. The article is well written and will have impact to the readers.