Lawson Surfaces

Minimal surfaces in the 3-sphere

The Lawson minimal surfaces ξ_g,1 in various stereographic projections.

Two projections of the Lawson minimal surfaces *ξ_g,1* with genus g=2, 3, 4, 5, 6.

The Lawson minimal surfaces *ξ_g,1* in S³ of genus g=2, 3, 4, 5. The surfaces are steregraphically projected from S³ to R³ with curvature lines.

Family of CMC surfaces of genus 2 in S³, starting at the Lawson surface (upper left). As the necks shrink as the lobes expand, the surfaces converge to a double covered minimal sphere as singular limit.

A family of CMC surfaces in S³ of genus 2. This stereographic projection exposes their Z₃ symmetry. Unlike family I, this family is not connected to the Lawson minimal surface *ξ_2,1*, The flow is conjectured to limit to a necklace of three CMC spheres (lower right) as the conformal type degenerates. }

Another stereographic projection of genus 2 CMC surfaces. In this view, the flow can be seen to mimic that of the 2-lobe Delaunay tori, but with a piece of a Delaunay cylinder glued in.

Genus 2 CMC surfaces with five and six lobes.

Lawson Surfaces

Minimal surfaces in the 3-sphere

The Lawson minimal surfaces ξ_g,1 in various stereographic projections.

Button view of the Lawson minimal surface ξ_2,1 cut along lines at 45 degrees to the curvature lines.

Lawson Surfaces

Minimal surfaces in the 3-sphere

The Lawson minimal surface ξ_g,1 of genus 2 with curvature lines rotated to closed curvature lines with rational slope.

The Lawson minimal surface ξ_2,1 of genus 2 with rotated curvature lines of slope 1:2.

Button view of the Lawson minimal surface ξ_2,1 of genus 2 with rotated curvature lines of slope 1:2.

Lawson minimal surface ξ_2,1 of genus 2 with rotated curvature lines of slope 4:5.

Cutaway view of Lawson minimal surface ξ_2,1 of genus 2 with rotated curvature lines of slope 4:5.

Lawson Surface of genus 4

Minimal surfaces in the 3-sphere

Lawson minimal surface ξ₂₂ of genus 4.

“Standard view” of the Lawson ξ_2,2 minimal surface. An order 3 symmetry is apparent in this stereographic projection.

The surface is similar to the Lawson ξ_2,1 surface of genus 2, but with a trinoid inserted at the center which connects the three vertical columns.

“Button view” of the Lawson ξ_2,2 minimal surface, showing several order 2 symmetries.

The white lines on the surface are curvature lines, and the black lines divide the surface into 9 Plateau solutions.

Standard view, cut away by a geodesic 2-sphere.

One of the 9 isometric Plateau solutions which compose the surface. The Plateau solution is the minimal surface bounded by four edges of a geodesic tetrahedron which tiles S³. Since the Plateau solution is composed of 8 fundamental pieces, the order of the symmetry group of the surface is 72.

The conjugate cousin to the Lawson ξ_2,2 surface.

The conjugate cousin is a doubly periodic constant mean curvature in R³.

Lawson Symmetric Constant Mean Curvature Surfaces

CMC surfaces in the 3-sphere

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

Lawson symmetric CMC surface of genus 2.

Lawson Symmetric Constant Mean Curvature Surfaces

CMC surfaces in the 3-sphere

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

Flow through Lawson Surfaces

Minimal surfaces in the 3-sphere

Flow from the Clifford torus ξ_1,1 of genus 1 to the Lawson minimal surfaces ξ_1,2 of genus 2.

The flow breaks the topology.

The flow of the Plateaux solution from genus 1 to genus 2.

Flow through Lawson Surfaces

Minimal surfaces in the 3-sphere

The Lawson surfaces ξ_p,q are a family of minimal surfaces in S³ 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.