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Loop Weierstrass Representation

Thomas Raujouan, Nick Schmitt, Jonas Ziefle

Abstract

We introduce the Loop Weierstrass Representation for minimal surfaces in Euclidean space and constant mean curvature 1 surfaces in hyperbolic space by applying integral system methods to the Weierstrass and Bryant representations. We unify associated families, dual surfaces and Goursat transformations under the same holomorphic data, we introduce a simple factor dressing for minimal surfaces, and we compute and classify various examples.

Loop Weierstrass Representation

Abstract

We introduce the Loop Weierstrass Representation for minimal surfaces in Euclidean space and constant mean curvature 1 surfaces in hyperbolic space by applying integral system methods to the Weierstrass and Bryant representations. We unify associated families, dual surfaces and Goursat transformations under the same holomorphic data, we introduce a simple factor dressing for minimal surfaces, and we compute and classify various examples.

Paper Structure

This paper contains 33 sections, 47 theorems, 226 equations, 5 figures, 1 table.

Table of Contents

  1. The Loop Weierstrass Representation
  2. Ambient spaces
  3. Null curves and minimal or CMC $1$ immersions
  4. The Loop Weierstrass Representation
  5. Gauge freedom and standard forms of LWR potentials
  6. Spinors
  7. Weierstrass potential
  8. Schwarz potential
  9. Local description in the LWR
  10. First and second fundamental forms
  11. Gauss map
  12. Related surfaces
  13. Moving the evaluation points
  14. Associated family
  15. Dual associated family
  16. ...and 18 more sections

Key Result

Theorem 1.3

Let $\psi$ be a holomorphic null curve into $\mathrm{sl}_2\mathbb{C}$ . Then $\psi + \psi^*$ is a conformal, minimal immersion into Euclidean space. Moreover, any conformal, minimal immersion can locally be obtained this way.

Figures (5)

  • Figure 1: Simple factor dressing of the doubly-wrapped catenoid as in example \ref{['ex:dressed-catenoid1plane']} with $(p,q)=(1,1)$ and $(u,\ell)=(\frac{1}{2},1)$.
  • Figure 2: Five members in the dual associated family of a catenoid (definition \ref{['def:dual-assoc-family']}). The evaluation point $\lambda_0=0$ is fixed while $\lambda_1$ moves around the origin. All these immersions have the same hyperbolic Gauss map.
  • Figure 3: Simple factor dressing of the catenoid in $\mathbb{E}^3$, as in example \ref{['ex:dressed-catenoid2']} with $(p,q)=(\frac{1}{4},1)$ and $(u,\ell) = (2,2)$.
  • Figure 4: Simple factor dressing of the catenoid in $\mathbb{H}^3$ viewed in the Poincaré ball model, as in example \ref{['ex:dressed-catenoid2']} with $(p,q)=(\frac{1}{4},-0.1)$ and $(u,\ell) = (2,2)$.
  • Figure 5: Simple factor dressing of the doubly-wrapped catenoid in $\mathbb{E}^3$, as in example \ref{['ex:dressed-catenoid1plane']} with $(p,q)=(1,1)$ and $(u,\ell)=(\frac{1}{2},1)$ (see also figure \ref{['fig:dressed-catenoids-R3']}).

Theorems & Definitions (124)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3: Weierstrass weierstrass
  • Theorem 1.4: Bryant bryant1987
  • Definition 1.5
  • Lemma 1.6
  • proof
  • Theorem 1.7
  • Definition 2.1
  • Lemma 2.2
  • ...and 114 more