Minimal surfaces of finite total curvature in $\mathbb{M}^2 \times \mathbb{R}$

Rafael Ponte
24/01/2019 , 15:30 | Sala 236.

Minimal surfaces with finite total curvature in three-dimensional spaces have been widely studied in the recent decades. A celebrated result in this subject states that, if $\Sigma \subset \mathbb{R}^3$ is a complete immersed minimal surface of finite total curvature, then it has finite conformal type. Moreover, its Weierstrass data can be extended meromorphically to the punctures and its total curvature is an integral multiple of $4 \pi$.

In this talk, the goal is to present some theorems concerning minimal surfaces in $\mathbb{M}^2 \times \mathbb{R}$ having finite total curvature, where $\mathbb{M}^2$ is a Hadamard manifold. We obtain analogous versions of classical results in Euclidean three-dimensional spaces. The main result gives a formula to compute the total curvature in terms of topological, geometrical and conformal data of the minimal surface. In particular, we prove the total curvature is an integral multiple of $2\pi$.