It follows from the famous Weierstrass representation, for MINIMAL surfaces in $3$-dimensional Euclidean space, that any such surface can locally be presented by a (weakly) CONFORMAL HARMONIC immersion from the complex plane given in terms of two holomorphic functions.

The inverse image of a regular value of a complex-valued holomorphic function on a Kähler manifold is a MINIMAL submanifold of codimension $2$. It is a direct consequence of the Cauchy-Riemann equations that such a function is a horizontally (weakly) CONFORMAL HARMONIC submersion.

Harmonic morphisms are maps $(M,g)\rightarrow (N,h)$ between Riemannian manifolds generalizing holomorphic functions from Kähler manifolds. If the codomain is the complex plane any regular fibre is a MINIMAL submanifold of $M$ of codimension $2$.

In this talk we will give a brief introduction to the theory of harmonic morphisms and then discuss the existence of complex-valued solutions from Lie groups and symmetric spaces.

In this talk, we will prove an Alexandrov type theorem for a quotient space of $\mathbb{H}^2 \times \mathbb{R}$. More precisely, we will classify the compact embedded surfaces with constant mean curvature in the quotient of $\mathbb{H}^2 \times \mathbb{R}$ by a subgroup of isometries generated by a parabolic translation along horocycles of $\mathbb{H}^2$ and a vertical translation. Moreover, we will construct some examples of periodic minimal surfaces in $\mathbb{H}^2 \times \mathbb{R}$ .

In this talk we will consider min-max minimal surfaces in three-manifolds and describe some related scalar curvature rigidity results. We will also mention some sharp estimates for the width in the case of positive Ricci curvature. The proofs use Ricci flow. This is joint work with Andre Neves.

In previous work with Spyros Alexakis, we considered the renormalized energy of complete properly embedded minimal surfaces in $\mathbb{H}^3$ and proved several structure theorems about it. I will report on that older work as well as our new results showing how control on this renormalized area yields a certain amount of regularity of the asymptotic boundary at infinity.