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Helicoidal minimal surfaces of prescribed genus, I

David Hoffman, Martin Traizet, Brian White

Abstract

For every genus g, we prove that S^2 x R contains complete, properly embedded, genus-g minimal surfaces whose two ends are asymptotic to helicoids of any prescribed pitch. We also show that as the radius of the S^2 tends to infinity, these examples converge smoothly to complete, properly embedded minimal surfaces in Euclidean 3-space R^3 that are helicoidal at infinity. In a companion paper, we prove that helicoidal surfaces in R^3 of every prescribed genus occur as such limits of examples in S^2 x R.

Helicoidal minimal surfaces of prescribed genus, I

Abstract

For every genus g, we prove that S^2 x R contains complete, properly embedded, genus-g minimal surfaces whose two ends are asymptotic to helicoids of any prescribed pitch. We also show that as the radius of the S^2 tends to infinity, these examples converge smoothly to complete, properly embedded minimal surfaces in Euclidean 3-space R^3 that are helicoidal at infinity. In a companion paper, we prove that helicoidal surfaces in R^3 of every prescribed genus occur as such limits of examples in S^2 x R.

Paper Structure

This paper contains 25 sections, 52 theorems, 142 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Helicoidal minimal surfaces in $\mathbf{S} ^2\times \mathbf{R}$
  3. The Main Theorems
  4. Positivity/Negativity of Surfaces at $O$
  5. $Y$-surfaces
  6. Periodic genus-$g$ helicoids in $\mathbf{S}^2\times\mathbf{R}$: theorem \ref{['theorem1']} for $h<\infty$
  7. Adjusting the pitch of the helicoid
  8. Eliminating jacobi fields by perturbing the metric
  9. Rounding the curve $\Gamma$ and the family of surfaces $t\mapsto S(t)$
  10. Roundings of $\Gamma$
  11. The existence of bridged approximations to $S$
  12. Additional properties of the family $t\mapsto S(t)$
  13. Counting the number of points in $Y\cap S(t)$
  14. Counting minimal surfaces bounded by smooth curves
  15. General results on existence of limits
  16. ...and 10 more sections

Key Result

Theorem 1

Let $H$ be a helicoid in $\mathbf{S}^2\times \mathbf{R}$ that has vertical axes $Z\cup Z^*$ and that contains the horizontal great circle $X$. For each genus $g\ge 1$ and each height $h\in (0,\infty]$, there exists a pair $M_{+}$ and $M_{-}$ of embedded minimal surfaces in $\mathbf{S}^2\times \mathb

Figures (5)

  • Figure 1: Right: A $Y$-surface of genus two. The number of fixed points of $\rho _Y$ ($180$-degree rotation around $Y$) is even (equal to six) and the number of boundary components is two. Center: A $Y$-surface of genus one. The number of fixed points of $\rho _Y$ is odd (equal to three) and there is a single boundary component. Left: This annular surface $A$ is not a $Y$-surface. The rotation $\rho_Y$ acts as the identity on $H_1(A,\mathbf{Z})$, not as multiplication by $-1$.
  • Figure 2: The boundary curve $\Gamma_C$. We depict $\mathbf{S}^2\times\mathbf{R}$ in these illustrations as $\mathbf{R}^3$ with each horizontal $\mathbf{S}^2\times\{z\}$ represented as horizontal plane via stereographic projection, with one point of the sphere at infinity. Here, that point is the antipodal point of the midpoint of the semicircle $Y^+$. Right: For ease of illustration, we have chosen the reference helicoid $H$ to be the vertical cylinder $X\times\mathbf{R}$, and the semicircle $C=C_\text{top}$ to meet $H$ orthogonally. The geodesics $X$, $Z$ and $Z^*$ divide $H$ into four components, two of which are shaded. The helicoid $H$ divides $\mathbf{S}^2\times\mathbf{R}$ into two components. The component $H^+$ is the interior of the solid cylinder bounded by $H$. Left: The boundary curve $\Gamma=\Gamma _C$ consists of the great circle $X$, two vertical line segments on the axes $Z\cup Z^*$ of height $2h$ and two semicircles in $(\mathbf{S}^2\times\{\pm h\})\cap H^+$. Note that $\Gamma$ has $\rho_Y$ symmetry. We seek a $\rho_Y$-invariant minimal surface in $H^+$ that has boundary $\Gamma_C$ and has all of its topology concentrated along $Y^+$. That is, we want a $Y$-surface as defined Section \ref{['section:$Y$-surfaces']} with the properties established in Proposition \ref{['Y-surface-topology-propostion']}. According to theorem \ref{['special-existence-theorem']}, there are in fact two such surfaces for every positive genus.
  • Figure 3: Rounding the corners of $\Gamma$.Center: The boundary curve $\Gamma$ as illustrated in Figure \ref{['GammaFigure']}. Left and Right: Desingularizations of $\Gamma$. The corners at $O$ and $O^*$ are removed, following the conditions $(1)$ and $(2)$ of \ref{['roundings-subsection']}. In both cases we have desingularized near $O$ by joining $X^+$ to $Z^+$ and $X^-$ to $Z^-$. In the language of Definition \ref{['PosNegRounding']}, both desingularizations are positive at $O$. On the left, the rounding is also positive at $O^*$. On the right, the rounding is negative at $O^*$. Note that when the signs of the rounding agree at $O$ and $O^*$, as they do on the left, the rounded curve has two components; when the signs are different, as on the right, the rounded curve is connected.
  • Figure 4: The sign of $S$ and $\Gamma (t)$ at $O$. The behavior near $O$ of a surface $S\subset H^+$ with boundary $\Gamma$. First Column: The surface $S$, here illustrated by the darker shading, is tangent at $O$ to either the positive quadrants of $H$ (as illustrated on top) or the negative quadrants (on the bottom). In the sense of section \ref{['sign-section']}, $S$ is positive at $O$ in the top illustration and negative in the bottom illustration. Second column: A curve $\Gamma (t)$ in a positive rounding $t\rightarrow \Gamma(t)$ of $\Gamma$. The striped regions lie in the projections $\Omega (t)$ defined in Theorem \ref{['bridged-approximations-theorem']}. Note that on the top $O\not\in \Omega (t)$. On the bottom, $O\in \Omega (t)$. Third Column: A curve $\Gamma (t)$ in a negative rounding of $\Gamma$. The striped regions lie in $\Omega (t)$. Note that on top we have $O\in \Omega (t)$. On the bottom, $O\not \in \Omega(t)$.
  • Figure 5: Left: the shaded region is the intersection of $H^+$ with the vertical cylinder of axis $Z$ and radius $r$. Right: intersection of $M$ with the same cylinder, unrolled in the plane. We use cylindrical coordinates $(r,\theta,z)$, with $\theta=0$ being the positive $Y$-axis. The radius $r$ is chosen so that the cylinder intersects a catenoidal neck. The positive $X$-axis intersects the cylinder at the point $(\theta,z)=(-\pi/2,0)$. The negative $X$-axis intersects the cylinder at the point $(\theta,z)=(\pi/2,0)$, which is the same as the point $(-3\pi/2,0)$ on the cylinder.

Theorems & Definitions (111)

  • Theorem 1
  • Remark 3.1
  • Remark 3.2
  • Theorem 2
  • Corollary
  • Theorem 3
  • Remark 4.1
  • Definition 5.1
  • Proposition 5.2
  • Corollary 5.3
  • ...and 101 more