# Lawson Surfaces

## Minimal surfaces in the 3-sphere

The Lawson minimal surfaces *ξ _{g,1}* in various stereographic projections.

The Lawson minimal surfaces *ξ _{g,1}* in various stereographic projections.

The Lawson minimal surfaces *ξ _{g,1}* in various stereographic projections.

The Lawson minimal surface *ξ _{g,1}* of genus 2 with curvature lines rotated to closed
curvature lines with rational slope.

Lawson minimal surface ξ_{22} of genus 4.

Family 0 of Lawson symmetric constant mean curvature surfaces of genus 2.

Family 1 of Lawson symmetric constant mean curvature surfaces of genus 2.

Flow from the Clifford torus *ξ _{1,1}* of genus 1 to the Lawson minimal surfaces

The Lawson surfaces ξ_{p,q} are a family of minimal
surfaces in S^{3} of
genus *pq*. There is a flow through minimal surface in which the two integers are replaced by real
parameters.

Shown below is one leg of this flow, from the Lawson surface
ξ_{2,1} of genus 2 to
ξ_{2,2} of genus 4. An order 2 symmetry of the initial surface with six fixed points is
“opened” along three cuts until it reaches an order 3 symmetry. The final surface is show
before and after the missing piece is filled in.

Flow from a Delaunay torus to a Lawson CMC surface of genus 2.

Flow from a Delaunay torus to a Lawson CMC surface of genus 2.

Lawson CMC surface with six lobes.

Surfaces with the symmetries of Platonic solids and tesselations of the 3-sphereÂ [1]. The surfaces in the WebGL, with genus *g*, are

- tetrahedron
*g*=3 - octahedron
*g*=7 - cube
*g*=5 - octahedral join
*g*=11 - icosahedron
*g*=19 - dodecahedron
*g*=11 - icosahedron join
*g*=29

- New minimal surfaces in
*S*, J. Differential Geom. 28 (1988), no. 2, 169—185 [961512].^{3}
,

These surfaces are build by putting tubes on regular tessellations of S^{3}Â [1]. The surfaces in the WebGL, with genus *g*, are

- 5-cell
*g*=6 - 16-cell
*g*=17 - 24-cell
*g*=73 - 600-cell
*g*=601

- New minimal surfaces in
*S*, J. Differential Geom. 28 (1988), no. 2, 169—185 [961512].^{3}
,

Torus with eight Delaunay ends.