DGD - Discretization in Geometry and Dynamics

$Fig. 3: Button view of the Lawson minimal surface $\xi_2,1$ cut along lines at 45 degrees to the curvature lines.$
$Fig. 4: Button view of the Lawson minimal surface $\xi_2,1$ of genus 2 with rotated curvature lines of slope 1-2.$
$Fig. 5: Cutaway view of Lawson minimal surface $\xi_2,1$ of genus 2 with rotated curvature lines of slope 4-5.$
$Fig. 8: Lawson minimal surface $\xi_2,1$ of genus 2 with rotated curvature lines of slope 4-5.$
$Fig. 32: The conjugate cousin is a doubly periodic constant mean curvature in $\mathbb{R}^3$.$
$Fig. 33: The conjugate cousin to the Lawson $\xi_{2,2}$ surface.$
$Fig. 34: The Lawson minimal surface $\xi_{2,1}$ of genus 2 with rotated curvature lines of slope 1-2.$
$Fig. 35: The Lawson minimal surface $\xi_{2,1}$.$
$Fig. 36: The Lawson minimal surface $\xi_{2,1}$.$
$Fig. 37: The Lawson minimal surface $\xi_{6,1}$.$
$Fig. 38: The Lawson minimal surfaces $\xi_{g,1}$ in $S^3$ of genus g=2, 3, 4, 5. The surfaces are steregraphically projected from $S^3$ to $\mathbb{R}^3$ with curvature lines.$
$Fig. 39: The surface is similar to the Lawson $\xi_{2,1}$ surface of genus 2, but with a trinoid inserted at the center which connects the three vertical columns.$
$Fig. 41: Two projections of the Lawson minimal surfaces $\xi_{g,1}$ with genus g=2, 3, 4, 5, 6.$
$Fig. 42: “Button view” of the Lawson $\xi_{2,2}$ minimal surface, showing several order 2 symmetries.$
$Fig. 43: “Standard view” of the Lawson $\xi_{2,2}$ minimal surface. An order 3 symmetry is apparent in this stereographic projection.$

Media

WebGL models

Description

In 1970, Herbert Blaine Lawson gave a description of complete minimal surfaces in $S^3$ [1]. For visualization one can use the stereographic projection to $\mathbb{R}^3$ as for instance is seen here. The resulting surfaces are known to be CMC-surface in euclidean 3-space. One can obtain Lawson surfaces by solving a Plateau problem on a spherical geodesic quadrilateral followed by suitable reflections.

We have images of:
- The Lawson minimal surfaces $\xi_{g,1}$ in various stereographic projections.
- A Lawson CMC surface with six lobes
- The Lawson minimal surface $\xi_{g,1}$ of genus 2 with curvature lines rotated to closed curvature lines with rational slope
- The family 0 of Lawson symmetric constant mean curvature surfaces of genus 2.
- The family 1 of Lawson symmetric constant mean curvature surfaces of genus 2.

References

Herbert B. Lawson.
Complete minimal surfaces in $S^3$.
The Annals of Mathematics, 1970.
URL: http://www.math.jhu.edu/~js/Math748/lawson.s3.pdf.

Authors

Dr. Nicholas Schmitt +

University: TU Berlin
Website: http://page.math.tu-berlin.de/~schmitt/

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	Flow through Lawson Surfaces Nicholas Schmitt WebGL models images		Minimal n-Noids in Hyperbolic and Anti-de Sitter 3-Space Alexander I. Bobenko, Sebastian Heller, Nicholas Schmitt 3D models images
	Minimal Surfaces in the 3-Sphere with Symmetries Nicholas Schmitt WebGL models images		Koebe Polyhedra Stefan Sechelmann 3D models images

Keywords

Media

Lawson CMC Surfaces

Media

WebGL models

Description

References

Authors

Dr. Nicholas Schmitt +

See also

Flow through Lawson Surfaces

Minimal n-Noids in Hyperbolic and Anti-de Sitter 3-Space

Minimal Surfaces in the 3-Sphere with Symmetries

Koebe Polyhedra