Static elliptic minimal surfaces in $$\hbox {AdS}_4$$ AdS 4

Abstract The Ryu–Takayanagi conjecture connects the entanglement entropy in the boundary CFT to the area of open co-dimension two minimal surfaces in the bulk. Especially in $$\hbox {AdS}_4$$ AdS 4 , the latter are two-dimensional surfaces, and, thus, solutions of a Euclidean non-linear sigma model on a symmetric target space that can be reduced to an integrable system via Pohlmeyer reduction. In this work, we construct static minimal surfaces in $$\hbox {AdS}_4$$ AdS 4 that correspond to elliptic solutions of the reduced system, namely the cosh-Gordon equation, via the inversion of Pohlmeyer reduction. The constructed minimal surfaces comprise a two-parameter family of surfaces that include helicoids and catenoids in H $$^3$$ 3 as special limits. Minimal surfaces that correspond to identical boundary conditions are discovered within the constructed family of surfaces and the relevant geometric phase transitions are studied.