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

## TL;DR

This paper investigates minimal surfaces with finite total curvature in the product space of a Hadamard surface and the real line, providing a formula linking total curvature to topological and geometric data.

## Contribution

It introduces a formula to compute total curvature of such minimal surfaces, showing it is always an integral multiple of 2π, extending understanding of their geometric properties.

## Key findings

- Total curvature is an integral multiple of 2π.
- Derived a formula relating total curvature to topological and conformal data.
- Characterized minimal surfaces of finite total curvature in $\mathbb{M}^2 imes \mathbb{R}$.

## Abstract

The goal of this article is to study minimal surfaces in $\mathbb{M}^2 \times \mathbb{R}$ having finite total curvature, where $\mathbb{M}^2$ is a Hadamard manifold. 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 $2\pi$.