We investigate constrained Willmore immersions in $\mathbb{R}^3$ (critical points of the Willmore energy constrained to a fixed conformal class). They satisfy a Willmore equation "perturbed" on the right-hand side by a term involving geometric data as well as a holomorphic quadratic form playing the role of a Lagrange multiplier. We allow both geometric singularities (where the immersive nature of the immersion degenerates) and singularities in the Lagrange multiplier. We develop local asymptotic expansions for the immersion, its first and second derivatives in terms of residues computed as circulation integrals around the point singularities. We deduce explicit "point removability" conditions ensuring the smoothness of the immersion. The results apply in particular to Willmore immersions and to CMC immersions.

Amidst the great activity in the past several years concerning the study of complete minimal surfaces in homogeneous three-manifolds, the study of minimal surfaces in $\mathbb{H}^2\times\mathbb{R}$ has witnessed particular success. In this talk we will focus on complete minimal surfaces with finite total curvature in $\mathbb{H}^2\times\mathbb{R}$. We will present the construction of some important examples, the properties they have to satisfy and the known classification results. Finally, we will introduce non-proper complete minimal disks embedded in $\mathbb{H}^2\times\mathbb{R}$ which are invariant by a vertical translation and have finite total curvature in the quotient. They are a counterexample which shows that the Calabi-Yau conjectures do not hold for embedded minimal surfaces in $\mathbb{H}^2\times\mathbb{R}$.