Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

New minimal surface doublings of the Clifford torus and contributions to questions of Yau

Nikolaos Kapouleas, Peter McGrath

Abstract

The purpose of this article is three-fold. First, we apply a general theorem from our earlier work to produce many new minimal doublings of the Clifford Torus in the round three-sphere. This construction generalizes and unifies prior doubling constructions for the Clifford Torus, producing doublings with catenoidal bridges arranged along parallel copies of torus knots. Ketover has also constructed similar minimal surfaces by min-max methods as suggested by Pitts-Rubinstein, but his methods apply only to surfaces which are lifts of genus two surfaces in lens spaces, while ours are not constrained this way. Second, we use this family to prove a new, quadratic lower bound for the number of embedded minimal surfaces in $\mathbb{S}^3$ with prescribed genus. This improves upon bounds recently given by Ketover and Karpukhin-Kusner-McGrath-Stern, and contributes to a question of Yau about the structure of the space of minimal surfaces in $\mathbb{S}^3$ with fixed genus. Third, we verify Yau's conjecture for the first eigenvalue of minimal surfaces in $\mathbb{S}^3$ in the following cases. First, for all minimal surface doublings of the equatorial two-sphere constructible by our earlier general theorem. Second, for all the Clifford Torus doublings constructed in this article.

New minimal surface doublings of the Clifford torus and contributions to questions of Yau

Abstract

The purpose of this article is three-fold. First, we apply a general theorem from our earlier work to produce many new minimal doublings of the Clifford Torus in the round three-sphere. This construction generalizes and unifies prior doubling constructions for the Clifford Torus, producing doublings with catenoidal bridges arranged along parallel copies of torus knots. Ketover has also constructed similar minimal surfaces by min-max methods as suggested by Pitts-Rubinstein, but his methods apply only to surfaces which are lifts of genus two surfaces in lens spaces, while ours are not constrained this way. Second, we use this family to prove a new, quadratic lower bound for the number of embedded minimal surfaces in with prescribed genus. This improves upon bounds recently given by Ketover and Karpukhin-Kusner-McGrath-Stern, and contributes to a question of Yau about the structure of the space of minimal surfaces in with fixed genus. Third, we verify Yau's conjecture for the first eigenvalue of minimal surfaces in in the following cases. First, for all minimal surface doublings of the equatorial two-sphere constructible by our earlier general theorem. Second, for all the Clifford Torus doublings constructed in this article.

Paper Structure

This paper contains 6 sections, 26 theorems, 89 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and Background from LDG
  3. New Doublings of the Clifford Torus
  4. The first eigenspace for minimal doublings of $\mathbb{S}^2_{eq}$
  5. The first eigenspace for minimal doublings of the Clifford torus
  6. Eigenvalues of the Laplacian on Flat Tori

Key Result

Theorem 2.16

Assuming that ALDfam holds, there is a smooth closed embedded minimal surface $\breve{M}$ doubling $\Sigma$ in $N$ satisfying Moreover $\breve{M}$ has genus $2g_\Sigma-1+|L|$ (where $g_\Sigma$ is the genus of $\Sigma$) and its area $|\breve{M}|$ satisfies

Figures (2)

  • Figure 1: A depiction of a torus knot $\gamma$ and singular set $L$ as in \ref{['dL']} with $v = (2, 3)$, $k=1$ and $m = 6$. The points of $L$ are denoted by black circles, and points $p$ of $\mathbb{T} \setminus L$ for which the reflection $\underline{\mathsf{R}}_{p, \nu(p)}$ about $p$ preserves $L$ are denoted by white circles.
  • Figure 2: A depiction of a torus knot $\gamma$ and singular set $L$ as in \ref{['ELup']} with $v = (2, 3)$, $k=1$ and $m = 6$. The points of $L$ are denoted by black circles, and points $p$ of $\mathbb{T} \setminus L$ for which the reflection $\underline{\mathsf{R}}_{p, \nu(p)}$ about $p$ preserves $L$ are denoted by white circles.

Theorems & Definitions (68)

  • Definition 1.3
  • Definition 2.3: $A^\perp$ and reflections $\underline{\mathsf{R}}_A$
  • Definition 2.6
  • Definition 2.8: Spaces of affine functions
  • Definition 2.10: LD solutions
  • Theorem 2.16: Theorem A, LDG
  • Definition 3.1: Torus knots
  • Lemma 3.4: Properties of $\gamma$
  • proof
  • Lemma 3.8: Properties of ${\mathscr{G} }^L_{\mathrm{sym}}$
  • ...and 58 more