A family of higher genus complete minimal surfaces that includes the Costa-Hoffman-Meeks one

TL;DR

The paper develops a family of minimal surfaces in $\mathbb{R}^3$ with arbitrarily high genus and three ends by leveraging the Enneper–Weierstrass representation and elliptic-function data on genus-one and higher-genus Riemann surfaces. It constructs a one-parameter genus-one family $S_x$ with three ends, including the Costa surface at $|x|=e_1$ and two Enneper-type ends with a middle planar end for $|x|\neq e_1$, all possessing dihedral symmetry $\mathcal{D}(4)$. Building on this, it defines a broader family $\Sigma_{k,t}$ for each $k\ge1$, such that $|t|=1$ recovers the Costa–Hoffman–Meeks surfaces $M_k$, while $|t|<\sqrt{2\sqrt{k+1}-1}$ yields immersed genus-$k$ surfaces with $C_T=-4\pi(3k+2)$ and two Enneper ends plus one planar end; these are symmetric under $\mathcal{D}(2k+2)$ and decompose into $4(k+1)$ fundamental pieces. The work also connects these new surfaces to known limits, including Scherk–Enneper and Scherk’s fifth surface, thereby enriching the taxonomy of finite-total-curvature minimal surfaces with high genus and multiple ends.

Abstract

In this paper, we construct a one-parameter family of minimal surfaces in the Euclidean $3$-space of arbitrarily high genus and with three ends. Each member of this family is immersed, complete and with finite total curvature. Another interesting property is that the symmetry group of the genus $k$ surfaces $Σ_{k,x}$ is the dihedral group with $4(k+1)$ elements. Moreover, in particular, for $|x|=1$ we find the family of the Costa-Hoffman-Meeks embedded minimal surfaces, which have two catenoidal ends and a middle flat end.

Figures (6)

• Figure 1: Computer graphics of the genus one minimal surfaces $S_x^1$ for: (a) $x=0$, (b) $x=-1/2+e_1$, (c) $x=e_1$ (Costa surface) and (d) $x=1/2+e_1.$
• Figure 2: Computer graphics of the genus two minimal surfaces $\Sigma_{2,t}$ obtained for: (a) $t=0$, (b) $t=0.8$, (c) $t=1$ (Costa-Hoffman-Meeks surface) and (d) $t=1.2.$
• Figure 4: Computer graphics of the genus one minimal surfaces $S_x^2$ obtained for: (a) $x=-e_1$ and (b) $x=-e_1-0.5$.
• Figure 5: $\mathcal{F}$-sheets on $\overline{M}_k.$
• Figure 6: Fundamental piece.

Theorems & Definitions (16)

• Theorem A
• Theorem B
• Theorem 1: Enneper-Weierstrass representation
• Definition 1
• Proposition 1
• Proposition 2: Costa.1984
• Proposition 3: Hoffman.1985
• Remark 1
• Proposition 4: Karcher.1989, Wohlgemuth.1991
• Theorem 2: Schwarz Reflection Principle (see, for instance, Fujimoto.2013Karcher.1989)