Flows of constant mean curvature tori in the 3-sphere: The equivariant case

Abstract

We present a deformation for constant mean curvature tori in the 3-sphere. We show that the moduli space of equivariant constant mean curvature tori in the 3-sphere is connected, and we classify the minimal, the embedded, and the Alexandrov embedded tori therein. We conclude with an instability result.

Mean-convex Alexandrov embedded constant mean curvature tori in the 3-sphere

“…We make the following claim: If a cylinder is mean-convex Alexandrov embedded then m = n = ±1 (for this ensures that the surface is simply wrapped with respect to the rotational period). To prove this claim first note that any spectral genus zero cylinder is a covering of a homogeneous embedded torus [16]. The complement of this homogeneous embedded torus with respect to S 3 consists of two connected components D ± , both diffeomorphic to D × S 1 .…”

“…By multiplying ξ with positive numbers and by rotating λ → e iϕ λ we may achieve tr(ξ −1 ξ 0 ) = − 1 16 . These transformations induce a reparameterisation of the corresponding ζ.…”

“…For spectral genus zero, the polynomial a is equal to a(λ) = − 1 16 . In this case h = ln µ in Definition 3.6 (iii) is a meromorphic function with simple poles at λ = 0, λ = ∞:…”

Towards a Constrained Willmore Conjecture

“…This result had been conjectured by Pinkall and Sterling [62] as a generalization of Lawson's conjecture. Embeddedness restricts those surfaces to be the products of circles, the homogeneous tori S 1 × S 1 (b), for any b ≥ 1, and the unduloidal k-lobed Delaunay tori [39] in a 3-sphere for k ≥ 2 (see Figure 2). The conformal types R 2 /Z ⊕ τ Z of those tori sweep out all rectangular conformal types τ = ib for b ∈ [1, ∞) starting at the square structure.…”

Whitham deformations and the space of harmonic tori in S3

“…The form of Whitham deformations used here resemble their application in the theory of constant mean curvature surfaces, with similar goals. In it was shown that the space of equivariant CMC tori in is a connected infinite graph. uses Whitham deformations to show that for each and each fixed genus of the spectral curve, spectral triples of tori of constant mean curvature in are dense amongst those of CMC planes.…”