[Paper Review] On symmetries of constant mean curvature surfaces

Josef F. Dorfmeister, Guido Haak|ArXiv.org|Mar 14, 1996

Geometric Analysis and Curvature Flows14 references19 citations

TL;DR

This paper investigates the symmetries of conformal constant mean curvature (CMC) immersions from simply connected Riemann surfaces into R³, focusing on the interplay between automorphism groups of the universal cover, the Riemann surface, and spatial symmetries of the immersed surface. It establishes that spatial symmetries lift to biholomorphic automorphisms of the universal cover under specific conditions, particularly when the immersion arises from a meromorphic potential via the DPW method, and classifies such symmetries via the structure of the potential and monodromy.

ABSTRACT

We start the investigation of immersions $Ψ$ of a simply connected domain $D$ into three dimensional Euclidean space $R^3$, which have constant mean curvature (CMC-immersions), and allow for a group of automorphisms of $D$ which leave the image $Ψ(D)$ invariant. On one hand, this leads to a detailed description of symmetric CMC-surfaces and the associated symmetry groups. On the other hand, it allows us to start the classification of CMC-immersions of an arbitrary, compact or noncompact Riemann surface $M$ into $R^3$ in terms of Weierstrass-type data, as introduced by Pedit, Wu, and one of the authors [D]. We use our general results to prove, that there are no CMC-tori or Delaunay surfaces in the dressing orbit of the cylinder. As an example, we apply the discussion to Smyth surfaces and to a CMC-surface with a branchpoint.

Motivation & Objective

• To understand the structure of spatial symmetry groups AutΨ(D) for conformal CMC-immersions Ψ:D→R³.
• To relate spatial symmetries to biholomorphic automorphisms of the universal cover D and the underlying Riemann surface M.
• To characterize when spatial symmetries lift to automorphisms in AutΨ(D) using the DPW method and monodromy.
• To analyze examples with branch points and rotational symmetries, such as Smyth surfaces and surfaces with meromorphic potentials having poles or zeros.

Proposed method

• Uses the DPW method to represent CMC immersions via meromorphic connections and potentials on the universal cover D.
• Analyzes transformation properties of the metric and Hopf differential under automorphisms in AutΨ(D), deriving necessary conditions for symmetry lifting.
• Applies the monodromy representation to determine when a biholomorphic automorphism of D lifts to a spatial isometry of the image surface.
• Considers the universal cover C and constructs explicit potentials for surfaces with rotational symmetry, such as those with a single umbilic of order m.
• Studies surfaces with branch points by removing the branch locus and analyzing the induced potential on the punctured cover.
• Uses coordinate transformations (e.g., w = e^w̃) to express the potential in terms of the universal cover and determine the elementary group.

Experimental results

Research questions

• RQ1 Under what conditions does a biholomorphic automorphism of the universal cover D lift to a spatial isometry of the CMC surface Ψ(D)?
• RQ2 How do the automorphism groups AutM, AutD, and AutΨ(D) relate for a CMC-immersion Φ:M→R³?
• RQ3 What is the role of the monodromy representation in determining spatial symmetries of CMC surfaces?
• RQ4 How do meromorphic potentials with zeros or poles relate to the existence of spatial symmetries, such as rotational or translational invariance?
• RQ5 What is the structure of the elementary group for CMC surfaces with branch points, and how does it affect the potential on the universal cover?

Key findings

• A nondegenerate Smyth surface arises if and only if m=0 and |c|≠1, where m is the order of the umbilic and c is a parameter in the potential.
• The potential ξ=λ⁻¹((0 z−z₀; 1 0))dz on C gives a nonsingular CMC immersion with one branchpoint, even though f(z)=z−z₀ is not a square of a meromorphic function.
• On the universal cover, the potential transforms to ξ̃=λ⁻¹((0 e²w̃; e^w̃ 0))dw̃, and the elementary group is generated by translation w̃↦w̃+2πi.
• The monodromy of the potential determines whether a biholomorphic automorphism of D lifts to a spatial isometry, and this is governed by the behavior of the potential under the automorphism.
• For surfaces with branchpoints, the correct potential on the punctured cover is obtained by pulling back via the exponential map, and the resulting monodromy encodes the symmetry group.
• The group AutΨ(D) is isomorphic to the subgroup of AutD that preserves the potential and lifts to a spatial isometry, linking complex-analytic and geometric symmetries.