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GENUS
$1$ MINIMAL k-NOIDS AND SADDLE TOWERS IN 
$\mathbb {H}^2\times \mathbb {R}$
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 22 / Issue 5 / September 2023
- Published online by Cambridge University Press:
- 06 January 2022, pp. 2155-2175
- Print publication:
- September 2023
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- Article
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For each
$k\geq 3$, we construct a 
$1$-parameter family of complete properly Alexandrov-embedded minimal surfaces in the Riemannian product space 
$\mathbb {H}^2\times \mathbb {R}$ with genus 
$1$ and k embedded ends asymptotic to vertical planes. We also obtain complete minimal surfaces with genus 
$1$ and 
$2k$ ends in the quotient of 
$\mathbb {H}^2\times \mathbb {R}$ by an arbitrary vertical translation. They all have dihedral symmetry with respect to k vertical planes, as well as finite total curvature 
$-4k\pi $. Finally, we provide examples of complete properly Alexandrov-embedded minimal surfaces with finite total curvature with genus 
$1$ in quotients of 
$\mathbb {H}^2\times \mathbb {R}$ by the action of a hyperbolic or parabolic translation.
