Gluing Karcher–Scherk saddle towers I: Triply periodic minimal surfaces

Lemma A.1.

We have H ˙ = 0 on the Riemann sphere minus the points p h ∘ for h ∈ H .

Proof.

We have for all ϵ

We have λ i ∘ = 0 and λ ˙ i = D ⁢ Λ i ⁢ ( p ∘ ) ⋅ p ˙ = 0 so Q ˙ = 0 . Let w = x 1 + i ⁢ x 2 be a local complex coordinate on the Riemann sphere. Recall the standard formula for the mean curvature

where g and b are respectively the matrices of the first and second fundamental forms in the coordinate system ( x 1 , x 2 ) . We have for all ϵ

Hence from Q ∘ = 0 and Q ˙ = 0 we obtain

Since X ⁢ ( ϵ ) is harmonic for all ϵ, we have b 11 + b 22 = 0 for all ϵ, so

Taking the derivative of (A.1), we obtain H ˙ = 0 . ∎

Lemma A.2.

The map ξ ⁢ d d ⁢ w defines a holomorphic vector field on the Riemann sphere. Moreover, ξ ⁢ ( p h ∘ ) = - p ˙ h for h ∈ H .

Proof.

Since X ∘ is conformal and harmonic, we have

where 〈 ⋅ , ⋅ 〉 denotes the ℂ -bilinear dot product. For all ϵ

Taking the derivative with respect to ϵ and using equation (A.2), we obtain

Hence ∂ ⁡ ξ ¯ ∂ ⁡ w = 0 and ξ is holomorphic. If w ′ is another local complex coordinate, we can write

This gives

so ξ transforms as a holomorphic vector field under change of coordinate. Using the complex coordinate w = z - p h ∘ in a neighborhood of p h ∘ , we have

Hence by equation (A.3),

Proposition B.1.

The graph is balanced if and only if x ∘ is a critical point of L restricted to those x ∈ A such that P hor ⁢ ( x ) = P hor ⁢ ( x ∘ ) .

Theorem C.1.

The Meeks, aG, aH, and aI configurations in Figure 6 are the only balanced configurations whose graphs are orientable with two faces. Hence they are the only possible configurations that give rise to TPMSs of genus 3.

Proof.

If a configuration gives rise to a TPMS of genus three, its graph must have two faces. By Euler’s formula, the average degree of the graph is

By Assumption 3.6, the average degree is at least 4, hence the graph has at most 2 vertices. We discuss two cases

Case | V | = 1 , | E | = 3 . All edges are loops. Since edges are represented by straight segments, no loop is null-homologous. If two loops are homologous, they must be parallel, and represented by the same segment that cuts the torus into an annulus. To form a 2-cell embedding, the remaining edge must be in a different homology class. But then, it is impossible to orient the half-edges alternately incoming and outgoing around the vertex, contradicting the orientability.

So we have three pairwise non-homologous simple loops. They only intersect at the vertex, hence any two of them form a homology basis, and the remaining loop must be homologous to their concatenation. So the graph must be homeomorphic to that of the aH.

Any configuration with this graph is trivially balanced because all edges are loops. It is also trivially rigid as there is only one vertex, so the cut space is trivial.

Case | V | = 2 , | E | = 4 . We first prove that such a graph has no loop. The graph is connected, hence the edges can not be all loops. If exactly one or three edges are loops, the degree of a vertex will be smaller than 4, contradicting Assumption 3.6. If exactly two edges are loops, they must be adjacent to different vertices. So they divide the torus into two annuli. Since the graph is represented as a limit of 2-cell embeddings, the remaining edges must lie in different annuli. But then, it is impossible to orient the half-edges alternately incoming and outgoing around each vertex, contradicting the orientability. This proves that none of the edges is a loop.

So we have four edges between two vertices. Any two of the edges form a cycle. If some of these cycles are null-homologous, we will have parallel edges. If at most one edge is simple, it is not possible to form a 2-cell embedding. If exactly two edges are simple, the only 2-cell embedding does not have a proper orientation.

So we have four simple edges between two vertices. Then the graph must be as shown in Figure 11. This graph is balanced if and only if the half-edges form two collinear pairs around each vertex. So if one vertex is at 0, the other vertex must lie at a 2-division point.

By the period condition (6.1), the phase function must be of the form

on the half-edges around a vertex, where Ψ 1 and Ψ 2 are the fundamental shifts. See Figure 11.

If arg ⁡ ( T 2 T 1 ) ≠ π 2 , then the shortest edges of the graph form a collinear pair. By assuming arg ⁡ ( T 2 T 1 ) < π 2 as in Figure 11, the phase function is balanced if and only if

The solution Ψ 1 = - 2 ⁢ c gives Meeks’ configuration which is generically rigid. The solution Ψ 2 - Ψ 1 = π gives the aI configurations, which is not rigid because c remains a free variable.

If arg ⁡ ( T 2 T 1 ) = π 2 , the phase function is balanced if and only if

The solution Ψ 1 = - 2 ⁢ c gives again Meeks’ configurations. The solutions Ψ 1 = π and Ψ 2 = π give the aG configurations, which is not rigid because c remains a free variable. ∎

Theorem D.1.

Let ω t be a family of holomorphic 1-forms on A t , depending holomorphically on t ∈ D * ⁢ ( 0 , ϵ 2 ) , and let ω ~ t = ψ t * ⁢ ω t . Define

Assume that

where ω 0 and ω ~ 0 are holomorphic in D * ⁢ ( 0 , ϵ ) with at most simple poles at z = 0 , and the limit is uniform on compact subsets of D * ⁢ ( 0 , ϵ ) . Then ∫ β t ω t - α t ⁢ log ⁡ t is a well-defined holomorphic function of t ≠ 0 which extends holomorphically at t = 0 . Moreover, its value at t = 0 is

Proof.

If arg ⁡ ( t ) is increased by 2 ⁢ π , the homotopy class of β t is multiplied on the left by γ, so ∫ β t ω t is increased by 2 ⁢ π ⁢ i ⁢ α t . On the other hand, log ⁡ t is increased by 2 ⁢ π ⁢ i , so the difference ∫ β t ω t - α t ⁢ log ⁡ t is a well-defined holomorphic function of t in D * ⁢ ( 0 , ϵ 2 ) . Using the change of variable rule, we write

To estimate the first term, we fix ϵ 1 < ϵ 2 < ϵ and expand ω t in Laurent series in the annulus 𝒜 t as

with a 0 ⁢ ( t ) = α t and

Since ω t and ω ~ t are uniformly bounded on the circle | z | = ϵ 2 , this gives the estimates, for n > 0 and a uniform constant C,

Then we have

Using estimates (D.2) and ϵ 1 < ϵ 2 , it is straightforward to check that the sum is uniformly bounded with respect to t and that we have

On the other hand, we have

so

The second term in (D.1) is estimated in the exact same way, leading to

The function ∫ β t ω t - α t ⁢ log ⁡ t is bounded so extends holomorphically at t = 0 by the Riemann Extension Theorem.

The last point of Theorem D.1 follows from equations (D.1), (D.3) and (D.4). ∎

Proof of Lemma 8.3.

Recall the definition of the path B h just before Lemma 8.3. On path number (1), ω t depends holomorphically on t h in a neighborhood of 0 so

is a holomorphic function of t h in a neighborhood of 0. Same for path number (3). Regarding path number (2) we write

and we apply Theorem D.1 to the 1-form ( w h - 1 ) * ⁢ ω t with ϵ 1 = δ , t = t h , observing that

so the hypotheses of Theorem D.1 are satisfied. Hence (D.5) extends holomorphically at t h = 0 and its value there is

Adding the three terms gives Lemma 8.3. ∎