Moduli Space of Harmonic Tori in S 3 Ross Ogilvie School of Mathematics and Statistics University of Sydney June 2017 1 / 23
Harmonic Map ◮ Let f : T 2 → S 3 = SU (2) be a harmonic map. ◮ A harmonic map is a critical point of the energy functional. ◮ Long historical interest in minimal and constant curvature surfaces. A surface is CMC iff its Gauss map is harmonic. ◮ Minimal surfaces = conformal harmonic = CMC with zero mean curvature. ◮ Thought to be quite rare; Hopf Conjecture. Wente (1984) constructed immersed CMC tori. ◮ A classification of such maps is given by spectral data (Σ , Θ , ˜ Θ , E ) (Hitchin, Pinkall-Sterling, Bobenko). 2 / 23
Spectral Data (Σ , Θ , ˜ Θ , E ) ◮ Spectral curve Σ is a real (possibly singular) hyperelliptic curve, η 2 = � ( ζ − α i )(1 − ¯ α i ζ ) ◮ Θ , ˜ Θ are differentials with double poles and no residues over ζ = 0 , ∞ . ◮ Period conditions: The periods of Θ , ˜ Θ must lie in 2 π i Z . ◮ Closing conditions: for γ + a path in Σ between the two points over ζ = 1, and γ − between the points over ζ = − 1 then � � � � ˜ ˜ Θ , Θ , Θ , Θ ∈ 2 π i Z . γ + γ − γ + γ − ◮ E is a quaternionic line bundle of a certain degree. 3 / 23
CMC Moduli Space (Kilian-Schmidt-Schmitt) ◮ One can vary the line bundle E , so called isospectral deformations. ◮ CMC non-isospectral deformations. Maps come in one dimensional families. ◮ M CMC is disjoint lines parametrised by H ∈ R 0 ◮ Components M CMC end in either M CMC or bouquet of spheres. 1 0 4 / 23
Harmonic Map Example ◮ f ( x + iy ) = exp( − 4 x X ) exp(4 y Y ), for � 0 � 0 � � 1 δ X = , Y = , I m δ > 0 − 1 0 − δ 0 ◮ This map is periodic. Formula well-defined on any torus C / Γ, where Γ is a sublattice of this periodicity lattice. Im z π 4 | δ | Re z π 4 5 / 23
◮ Holding either x or y constant gives circles. ◮ As δ → R × , image collapses to a circle. ◮ As δ → 0 , ∞ , the periodicity lattice degenerates. 6 / 23
Constructing Spectral Data ◮ Up to translations, f is determined by the Lie algebra valued map f − 1 df , the pullback of the Mauer-Cartan form. ◮ Decompose into its dz and d ¯ z parts f − 1 df = 2(Φ − Φ ∗ ). ◮ Use f to pull pack the Levi-Civita connection on SU (2) to get a connection A . ◮ Given a pair (Φ , A ), we can make a family of flat SL (2 , C ) connections. Let ζ ∈ C × be the spectral parameter and define d ζ := d A + ζ − 1 Φ − ζ Φ ∗ Family of connections is ( X − i Y ) + ζ − 1 ( X + i Y ) � � d ζ = d − dz − [( X + i Y ) + ζ ( X − i Y )] d ¯ z = d − ζ − 1 [( X + i Y ) + ζ ( X − i Y )] [ dz + ζ d ¯ z ] 7 / 23
Holonomy ◮ Because the connections are flat, we can define holonomy for them. ◮ Pick a base point and generators for the fundamental group, ie take two loops around the torus. ◮ Parallel translating vectors with d ζ around one loop gives a linear map on the tangent space at the base point. Call this H ( ζ ). Around the other loop call the transformation ˜ H ( ζ ). ζ − 1 [( X + i Y ) + ζ ( X − i Y )] [ τ + ζ ¯ � � H τ ( ζ ) = exp τ ] 8 / 23
Spectral curve ◮ The fundamental group of T 2 is abelian, so H and ˜ H commute. Therefore they have common eigenspaces. ◮ Define ( ζ, L ) ∈ C × × C P 1 | L is an eigenline for H ( ζ ) � � Σ = closure ◮ The eigenvalues of H ( ζ ) are µ ( ζ ) , µ ( ζ ) − 1 . The characteristic polynomial is µ 2 − (tr H ) µ + 1 = 0 ◮ Using the compactness of the torus, one can show that (tr H ) 2 − 4 vanishes to odd order only finitely many times. The spectral curve is always finite genus for harmonic maps T 2 → S 3 . 9 / 23
◮ From example �� � ��� � − (1 + i ¯ � Σ = ζ, ± (1 − i δ )( ζ − α ) : δ )(1 − ¯ αζ ) for α = 1 + i δ δ = i 1 + α ⇔ − 1 + i δ 1 − α ◮ Can write equation for Σ as η 2 = ( ζ − α )(1 − ¯ αζ ) 10 / 23
The Differentials µ ( ζ ) of H ( ζ ) , ˜ ◮ The differentials come from the eigenvalues µ ( ζ ) , ˜ H ( ζ ). These functions have essential singularities. µ are holomorphic on C × and have simple poles ◮ However log µ, log ˜ above ζ = 0 , ∞ . ◮ d log µ removes the additive ambiguity of log. Thus we set Θ = d log µ and ˜ Θ = d log ˜ µ ◮ In order to recover the eigenvalues, one requires residue free double poles over ζ = 0 , ∞ and that the periods of the differentials lie in 2 π i Z . 11 / 23
◮ The eigenvalues of H τ ( ζ ) are τζ ) ηζ − 1 � � µ τ ( ζ, η ) = exp i | 1 − i δ | ( τ + ¯ . ◮ The corresponding differential is therefore τζ ) ηζ − 1 � � Θ τ = i | 1 − i δ | d ( τ + ¯ . ◮ On any given spectral curve, there is a lattice of differentials that may be used in spectral data. Different choices corresponds to coverings of the same image. 12 / 23
Moduli Space M 0 ◮ Every spectral curve in genus zero arises from this class of examples. ◮ Choice amounts to branch point α ∈ D 2 and choice of pair of differentials from a lattice � D 2 M 0 = α → S 1 \ {± 1 } . ◮ Image degenerates: δ → R × ⇔ ◮ Lattice degenerates: δ → 0 , ∞ ⇔ α → ± 1. ◮ Two dimensional (in contrast to CMC case). 13 / 23
Moduli Space M g Theorem At a point (Σ , Θ 1 , Θ 2 ) ∈ M g corresponding to a nonconformal harmonic map, if Σ is nonsingular, and Θ 1 and Θ 2 vanish simultaneously at most four times on Σ and never at a ramification point of Σ , then M g is a two-dimensional manifold in a neighbourhood of this point. Theorem At a point (Σ , Θ 1 , Θ 2 ) ∈ M g corresponding to a conformal harmonic map, if Σ is nonsingular, and Θ 1 and Θ 2 never vanish simultaneously on Σ then M g is a two-dimensional manifold in a neighbourhood of this point. ◮ Proof uses Whitham deformations. 14 / 23
Genus One − 1 . Let ◮ Spectral curves have two pairs of branch points α, β, α − 1 , β ( α, β ) ∈ D 2 × D 2 | α � = β � � A 1 = . ◮ Not every spectral curve has differentials that meet all the conditions. ◮ There is always an exact differential Θ E that meets all conditions except closing condition. ◮ A multiple of Θ E meets the closing condition if and only if S ( α, β ) := | 1 − α | | 1 − β | | 1 + α | | 1 + β | ∈ Q + 15 / 23
◮ Fix a value of p ∈ Q + . Let A 1 ( p ) = S − 1 ( p ). It is an open three-ball with a line removed. ◮ Rugby football shaped. Ends are ( α, β ) = (1 , − 1) , ( − 1 , 1). Seams are points with both α, β in S 1 . 16 / 23
◮ There is a second differential Θ P with periods 0 and 2 π i . Every differential that meets period conditions is a combination R Θ E + Z Θ P . ◮ Define T , up to periods of Θ P , by � � Θ P − Θ P 2 π iT := p γ − γ + ◮ A curve admits spectral data if and only if both S ∈ Q + and T ∈ Q (and the latter is well-defined). ◮ The connected components of the space of spectral curves are annuli if S = 1 and strips (0 , 1) × R if S � = 1. ◮ The connected components of the space of spectral data M 1 are all strips (0 , 1) × R . 17 / 23
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Method of Proof ◮ Move to the universal cover of the parameter space π p : ˜ A 1 ( p ) → A 1 ( p ) . ◮ Define a function ˜ T on ˜ A 1 ( p ) such that ˜ T = T ◦ π p . ◮ In the right coordinates, the level sets of ˜ T are graphs over (0 , 1) × R . ◮ Quotient by deck transformations to recover space of spectral curves. ◮ Consider how the lattice of differentials change as you change the spectral curve. 20 / 23
Interior Boundary M 1 ◮ M 1 ∩ A 1 ( p ) spirals around the diagonal line { α = β } ∩ A 1 ( p ). ◮ Just a single point on this diagonal line is reachable along a finite path. ◮ This limit seems not to be well-defined. 21 / 23
Exterior Boundary M 1 ◮ This boundary is where α or β tends to S 1 . ◮ A singular curve with a double point over the unit circle corresponds to genus zero spectral data via normalisation (blow-up). ◮ We can consider M 0 ⊂ ∂ M 1 . ◮ Each face of the football A 1 ( p ) is a disc, identified with the space of genus zero spectral curves. ◮ Edges of A 1 ( p ) correspond to all branch points on unit circle, ie a map to a circle. 22 / 23
Further questions ◮ Can we identify geometric properties that parameterise M ? ◮ Is M 0 ∪ M 1 connected? No. What other maps need to be included to make it connected? ◮ Can one deform a harmonic map to a circle to a harmonic map of any spectral degree? ◮ How does M g sit inside the moduli space of harmonic cylinders? Harmonic planes? ◮ What deformations lead to topological changes of the image of the harmonic map? 23 / 23
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