GENUS MINIMAL k-NOIDS AND SADDLE TOWERS IN

Jesús Castro-Infantes,José M Manzano

doi:10.1017/s1474748021000591

Jesús Castro-Infantes, José M Manzano

Open Access

https://doi.org/10.1017/s1474748021000591

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Abstract

AbstractFor 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.

Highlights

The theory of complete minimal surfaces in H2 × R with finite total curvature – that is, those whose Gauss curvature is integrable – has received considerable attention during the last decade, mainly triggered by Collin and Rosenberg [2]
The combined work of Hauswirth, Nelli, Sa Earp, and Toubiana [6], and Hauswirth, Menezes, and Rodrıguez [5] shows that a complete minimal surface immersed in H2 × R has finite total curvature if and only if it is proper, has finite topology, and has each of its ends asymptotic to an admissible polygon – that is, a curve homeomorphic to S1 consisting of finitely many alternating complete vertical and horizontal ideal geodesics
Ideal horizontal geodesics are those of the form Γ × {+∞} or Γ × {−∞}, where Γ is a geodesic of H2, whereas ideal vertical geodesics are those of the form {p∞} × R, where p∞ ∈ ∂∞H2 is an ideal point

Summary

Introduction

The theory of complete minimal surfaces in H2 × R with finite total curvature – that is, those whose Gauss curvature is integrable – has received considerable attention during the last decade, mainly triggered by Collin and Rosenberg [2]. Minimal k -noids constructed by Morabito and Rodrıguez [16] ( by Pyo [20] in the symmetric case) have finite total curvature, genus 0, and k ends asymptotic to vertical planes. There is a 2-parameter (resp., 1-parameter) family of properly embedded (resp., Alexandrov-embedded) minimal surfaces in H2 ×R with genus 0 and infinitely many ends, invariant by a discrete group of hyperbolic (resp., parabolic) translations Each of their ends is embedded and asymptotic to a vertical plane, and has finite total curvature. In the last part of the paper we will prove Theorem 3

Preliminaries

Construction of genus 1 saddle towers and k -noids

The conjugate construction

Solving the period problems

The embeddedness problem

Examples with infinitely many ends