Genus one minimal k-noids and saddle towers in $\mathbb{H}^2\times\mathbb{R}$

Abstract

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 $-4kπ$. Finally, we also 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.

Figures (8)

• Figure 1: Orientation of the conjugate surfaces $\Sigma$ and $\widetilde{\Sigma}$ according to the direction of rotation of $N$ along a vertical geodesic $\gamma$.
• Figure 2: Conjugate surfaces $\Sigma(a,\varphi,b)$ and $\widetilde{\Sigma}(a,\varphi,b)$ and their domains $\Delta$ and $\widetilde{\Delta}$ in $\mathbb H^2$ in the case $l=\infty$. Dashed lines represent ideal geodesics, and white dots represent ideal vertexes. The arrows in $\Sigma(a,\varphi,b)$ represent the normal $N$ at the endpoints of $v_2$ and $v_3$, which rotates counterclockwise along both geodesics.
• Figure 3: The angle $\theta_0$ of rotation of $\widetilde{v}_2$ with respect to the horocycle foliation at $\widetilde{v}_2(b)$, where we identify $\mathbb{H}^2\times\{0\}$ and $\mathbb H^2$. The surface $\widetilde{\Sigma}(a,\varphi,b)$ projects onto the shaded region $\widetilde{\Delta}$, with boundary the projections of the labeled curves. The complete geodesic $\gamma$ containing the projection of $\widetilde{h}_1$ appears in dotted line.
• Figure 4: On the left, boundary values for Jenkins--Serrin problems in $\mathbb{H}^2$ solved by $\Sigma(a,\varphi,b)$ and $\Sigma_{0}(b)$, where the perpendicular bisector of $\ell_1$ is represented in dotted line and $l<\infty$. On the right, the limit $\Sigma_\infty\subset\mathbb{R}^3$ by rescaling (fixing the length of $\ell_1$ equal to $1$) and the helicoid $\Sigma_0\subset\mathbb{R}^3$ in the proof of Lemma \ref{['lem:first-period']}.
• Figure 5: Tangent geodesics at $\widetilde{v}_2(0)$ and at a first $t_0\in(0,b]$ such that $\theta(t_0)=\pi$ (left). A first $t_0\in(0,b]$ such that $x(t_0)=0$ (center). A first $t_0\in(0,b]$ such that $\theta(t_0)=2\pi$ (right). The domains $U$ and $V$ are those where we apply Gauss--Bonnet formula in Lemma \ref{['lem:second-period']}.

Theorems & Definitions (18)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 1
• proof
• Lemma 2
• proof
• Remark 1
• Lemma 3
• proof