Minimal surfaces are one of the centre objects in Geometric Analysis, whose origins trace back to the 18th century. In 1744, Euler discovered the catenoid, one of the first known examples of a minimal surface. Since then, the theory has developed significantly, influencing various areas of mathematics. A fundamental question in minimal surface theory is: How can one construct new minimal surfaces?
Or even specifically, is it possible to construct or prove the existence of minimal surfaces of a given topological type? In this talk, we explore these questions. Starting with an exposition of Minimal surface theory, I will briefly explain three major directions in the construction of minimal surfaces: The Eigenvalue optimization methods (Spectral Theory), Min-Max constructions (Geometric Measure Theory) and Glueing constructions (PDE).
