Construction of complete hyperbolic minimal surfaces

Until the eighties, it was a general though that hyperbolic surfacesplayed a marginal role in the theory of minimal surfaces in Euclidean $3$-space. However the constructions of Jorge-Meeks in 1980 and Nadirashvili in 1996 have inspired the proof of general existence theorems for complete hyperbolic minimal surfaces. In this talk, we will discuss some techniques of construction of such surfaces and show several applications.