We construct embedded minimal surfaces of finite total curvature in euclidean space by gluing catenoids and planes. We use Weierstrass Representation and we solve the Period Problem using the Implicit Function Theorem. As a corollary, we obtain the existence of minimal surfaces with no symmetries.

: we prove the existence of embedded minimal surfaces of arbitrary genus g ≥ 3 in any flat 3-torus. In fact we construct a sequence of such surfaces converging to a planar foliation of the 3-torus. In particular, the area of the surface can be chosen arbitrarily large.

we classify the solutions to an overdetermined elliptic problem in the plane in the finite connectivity case. This is achieved by establishing a one-to-one correspondence between the solutions to this problem and a certain type of minimal surfaces.

AbstractWe construct constant mean curvature surfaces in euclidean space with genus zero and n ends asymptotic to Delaunay surfaces using the DPW method.