Minimal surfaces in the three-dimensional sphere

The three-sphere 𝕊³ is the three-dimensional boundary of a four-dimensional unit ball, much like the familiar two-sphere 𝕊² is the boundary of a three-dimensional unit ball. It is a classical setting for the study of minimal surfaces. In contrast to Euclidean space ℝ³, where minimal surfaces are either noncompact or with boundary, the three-sphere 𝕊³ contains a rich variety of closed minimal surfaces.

Visualizing any surface in 𝕊³ is challenging because 𝕊³ is a three-dimensional manifold embedded in four-dimensional space. Since ancient times, stereographic projection is a standard method for displaying spherical geometry in our Euclidean world. This mapping necessarily distorts distances and areas (just like any flat map of the Earth), but it preserves all angles. In particular, circles in the sphere are mapped to perfect circles or straight lines in Euclidean space. The figures displayed here are the images of minimal surfaces in 𝕊³ after stereographic projection to ℝ³. Regions of the surface that were near the pole of projection appear much larger than they really are, but in essence, we see a conformally faithful “shadow” of an object from 𝕊³.

References

• H. B. Lawson, Jr., Complete minimal surfaces in S³. Ann. of Math. (2) 92 (1970).
• M. B. Schulz and D. Wiygul, A new family of minimal surfaces of even genus in the three-dimensional sphere. Preprint (arXiv:2507.22531).
• H. Karcher, U. Pinkall, and I. Sterling, New minimal surfaces in S³. J. Differential Geom. 28 (1988).

The Lawson surfaces

The simplest minimal surfaces in 𝕊³ are the totally geodesic equatorial sphere and the flat Clifford torus. Beyond these, Lawson [1] constructed minimal surfaces ξ_m,k of genus m·k for any positive integers m,k. The surfaces ξ_m,k and ξ_k,m are congruent in 𝕊³. When k is large, ξ_m,k can be interpreted as a desingularisation of m+1 great spheres intersecting along a common equator at equal angles. In this sense, the Lawson surfaces are spherical analogues of the (later constructed) Karcher–Scherk towers in ℝ³ which desingularise m+1 planes intersecting along a common line.

Clifford torus ξ_1,1  
genus 1
area = 2π²

ξ_1,6   genus 6
area ≈ 7.595 π
symmetry order 112

ξ_2,6   genus 12
area ≈ 10.570 π
symmetry order 168

ξ_3,6   genus 18
area ≈ 13.006 π
symmetry order 224

ξ_4,4   genus 16
area ≈ 13.212 π
symmetry order 400

ξ_4,5   genus 20
area ≈ 14.219 π
symmetry order 240

ξ_4,6   genus 24
area ≈ 14.981 π
symmetry order 280

ξ_5,5   genus 25
area ≈ 15.540 π
symmetry order 576

ξ_5,6   genus 30
area ≈ 16.577 π
symmetry order 336

ξ_6,6   genus 36
area ≈ 17.871 π
symmetry order 784

New minimal surfaces of even genus in the 3-sphere

The following minimal surfaces in 𝕊³ have been discovered recently in [2] as spherical analogues of the celebrated Costa–Hoffman–Meeks surfaces in ℝ³ (in the same sense as the Lawson surfaces are spherical analogues of the Karcher–Scherk towers). In this analogy, the equatorial sphere plays the role of the plane and the Clifford torus plays the role of the complete catenoid. The surface of genus 4 in this family is conjectured to be congruent with the Lawson surface ξ_2,2, while the remaining surfaces are geometrically distinct from any other known example.

genus 12
symmetry order 48

genus 14
symmetry order 56

genus 24
symmetry order 96

The Karcher–Pinkall–Sterling surfaces

genus 11
 

genus 601
symmetry order 14400