Discrete constant mean curvature surfaces via conserved quantities inany space form
Wayne Rossman(joint work with F. Burstall, U. Hertrich-Jeromin, S. Santos)

Our purpose here is to present a definition for discrete constant mean curvature(CMC) h surfaces1 in any of the three space forms Euclidean 3-space R3 , spherical
3-space S 3 and hyperbolic 3-space H 3 . This new definition is equivalent to thepreviously known definitions [2] in the case of R3 . It also satisfies a Calapso
transformation relation (the Lawson correspondence), suggesting the definition isalso natural for the space form S 3 , and for CMC surfaces with h ≥ 1 in H 3 . The
definition is the first one for CMC surfaces with −1 < h < 1 in H 3 .
To motivate this definition for discrete CMC surfaces, we first consider the caseof smooth surfaces, and we begin by describing the 3-dimensional space forms
using the 5-dimensional Minkowski space R4,1 .
Minkowski 5-space. We give a 2 × 2 matrix formulation for Minkowski 5space. Let H denote the quaternions and Im H the imaginary quaternions.
½µ¶¯¾x x∞ ¯¯4,1R = X=x ∈ Im H, x0 , x∞ ∈ Rx0 −x ¯with signature (+, +, +, +, −) Minkowski metric hX, Y i such that hX, Y i · I =− 21 (XY + Y X), I = identity matrix. The 4-dimensional light cone is L4 = {X ∈R4,1 | ||X||2 = 0}. We can make the 3-dimensional space forms as follows: A space
form M is M = L4 ∩ {X | hX, Qi = −I} for any nonzero Q ∈ R4,1 . It turns outthat M has curvature κ, where Q2 = κ · I, so without loss of generality we canobtain any space form by choosingµ¶½µ¶¾
20 1x −x2
Q=, and then M = X =·,κ 01 −x1 − κx2
which is equivalent to {(x1 , x2 , x3 ) ∈ R3 ∪ {∞} |x21 + x22 + x23 6= −κ−1 }, wherex = x1 i + x2 j + x3 k ∈ Im H. Note that when κ < 0, M becomes two copies ofH 3 (κ).
Smooth surfaces in space forms. Letx = x1 (u, v)i + x2 (u, v)j + x3 (u, v)k ≈ X ∈ M
be a surface in M . Assume (u, v) is a conformal curvature-line coordinate system(every CMC surface can be parametrized this way). First we define the Christoffeltransformation x∗ , which for a CMC surface in R3 gives the parallel CMC surface:
Definition. The Christoffel transformation of x is any x∗ (defined in R3 up to−1translation) such that dx∗ = x−1
u du − xv dv.
In the next definition, the nonzero real constant c can be chosen freely:
12000 Mathematics Subject Classification: 53A10, 53C42, 37K35, 37K25
1µxdx∗Definition. For some c ∈ R \ {0}}, we set τ = cdx∗
4,1smooth Q and Z in R depending on (u, v) such that(1)¶−xdx∗ x. If there exist−dx∗ xd(Q + λZ) = (Q + λZ)λτ − λτ (Q + λZ)

holds for all λ ∈ R, then we call Q + λZ a linear conserved quantity of x.
Some properties of linear conserved quantities are immediate: Q and Z 2 areconstant; F τ = τ F = 0; F ⊥ Z and F ⊥ dZ. Properties like this can be utilizedto prove the following theorem:
Theorem. [1] The surface x is constant mean curvature in a space form M (produced by Q 6= 0) if and only if there exists (for that Q) a linear conserved quantity
Q + λZ.
Isothermic discrete surfaces and their Christoffel transforms. Consider
a discrete surface fp ∈ Im H, where p is any point in a discrete lattice domain.
Consider one quadrilateral in the lattice with vertices p, q, r, s ordered counterclockwise about the quadrilateral. We define the cross ratio of this quadrilateral
asqpqrs = (fq − fp )(fr − fq )−1 (fs − fr )(fp − fs )−1 .
When, for every quadrilateral, we can write the cross ratio asqpqrs = apq /aps ∈ R
so that the function apq defined on the edges of f satisfies
apq = asr ∈ R and aps = aqr ∈ R ,then we say that f is isothermic.
We can define the Christoffel transform f∗ of f bydf∗pq dfpq = apq .
We can then prove the following:
Lemma. [2] If f is isothermic, then there exists a discrete surface f∗ satisfying theabove equation for df∗ , and f∗ is isothermic with the same cross ratios as f.
Linear conserved quantities. We can now discretize (1) to obtain(2)(1 + λτpq )(Q + λZ)q = (Q + λZ)p (1 + λτpq ) ,where λ ∈ R and Q, Z ∈ R4,1 are functions on the lattice domain, andµ¶f df∗ −fp df∗pq fq
τpq = p ∗pq.
dfpq−df∗pq fq
We now come to the goal of this talk:Definition. If a linear conserved quantity Q+λZ, Q 6= 0, exists for an isothermic
discrete surface f, we say that f is of constant mean curvature in the space form
M determined by Q.
2Equation (2) can be extended to define polynomial conserved quantities.
In the figure, we show discrete CMC surfaces of revolution. The first two curves
are profile curves for discrete nonminimal CMC surfaces of revolution in R3 , thefirst being unduloidal and the second nodoidal. (For each of these two curves,the axis of rotation producing the surface is a vertical line drawn to the left ofthe curve, and is not shown in the figure.) The third picture shows the profilecurve for a discrete CMC surface of revolution in S 3 , where S 3 is stereographicallyprojected to R3 , and the shown circle is a geodesic of S 3 that is also the axis of thesurface – and furthermore, this example has a periodicity that causes it to closeon itself and form a torus. The final three pictures show discrete CMC surfaces ofrevolution in H 3 . The first two, with H > 1 and H = 1 respectively, are shown in
the Poincare model, and the first is unduloidal while the second looks similar to asmooth embedded catenoid cousin. (For these two curves, the corresponding axisof revolution is the vertical line between the uppermost and lowermost points ofthe circle shown, and this circle lies in the boundary sphere at infinity of H 3 .)The last picture is a minimal surface that lies in both copies of M = H 3 , andthe horizontal plane shown here is the virtual boundary at infinity of two copiesof the halfspace model for H 3 . This example was not known before, because thenotion of discrete CMC was not defined before in this case.
References[1] F. Burstall, D. Calderbank, Conformal submanifold geometry, in preparation.
[2] A. Bobenko, U. Pinkall, Discretization of surfaces and integrable systems, Oxford Lect. Ser.
Math. Appl. 16 (1999), 3–58.

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