A Note on the Hamiltonian Area Conjecture (Draft)

Tom Ilmanen^{1}

The aim of this note is to prove the following result.

Theorem. Let T be a smooth, immersed torus in C^{2}, Hamiltonian isotopic to the Clifford torus T_{c}. If T is stationary for area under Hamiltonian variations, then

|T | ≥ |T_{c}|,

where |T | denotes the area of T.

Supplemented by the existence and regularity of an area minimizer in this class, our result would imply the the 2-dimensional case of the Hamiltonian Area Conjecture [O1], which states that the Clifford torus in C^{n} minimizes area in its Hamiltonian isotopy class. Progress has been made toward such minimization problems in [SW].

We build on a result of Minicozzi [Mi], who showed that the above Theorem is a consequence of the Willmore Conjecture [W,Mi,T] in this Hamiltonian isotopy class. Our proof has three easy steps. First, we express the mean curvature form in the conformally equivalent flat metric on T, and find its (constant) length using Minicozzi’s observation. We apply the Isoperimetric Inequality to symplectic integrals to give a lower bound for the lengths of various curves in this flat metric. Finally we estimate the lattice in the universal cover to estimate the area of a fundamental domain, without, however, bounding the conformal type itself.

Preliminaries. We state a few definitions and facts about Lagrangian submanifolds and area. For more complete information see [O1,O2,Mo,Mi].

Let J denote the usual complex structure of C^{2} and ω the standard symplectic form, so ω(X, Y) = ⟨JX, Y ⟩. We identify forms with vectors using the metric whenever this is unambiguous. We use coordinates p = (x^{1} + iy^{1}, x^{2} + iy^{2}).

An immersed 2-surface T in C^{2} is called Lagrangian if ω|T_{p}T vanishes for each p ∈ T, or equivalently, if J(T_{p}T) = (T_{p}T)^{⊥}, so that

J : T_{p}T → (T_{p}T)^{⊥}

This document was prepared using AMS-TEX, August 1998. 1Max-Planck Institut für Mathematik, Inselstrasse 22-26, D-04103 Leipzig, Germany. Partially supported by NSF Grant DMS-9626405 and a Sloan Foundation Fellowship. defines an isometry.

A vector field of the form X = Jη, where η is a closed (resp. exact) 1-form defined on some open set, is called Lagrangian (resp. Hamiltonian. An isotopy T_{t} of immersed surfaces is called Lagrangian or Hamiltonian if its velocity vector has the corresponding property in an immersed tubular neighborhood of T_{t}. Note that if T is Lagrangian, the normal component of the motion X is determined solely by η pulled back to T.

The property of being Lagrangian is invariant under Lagrangian isotopy. We write T ∼ T′ if T is Hamiltonian isotopic to T′.

When T is Lagrangian, we define a 1-form h on T by rotating the mean curvature vector H into T_{p}T by the formula

h = −JH.

It follows from the Codazzi identity and the Lagrangian condition that

dh = 0,

so h represents a cohomology class [h] in H^{1}(T), called the Maslov class. This class is invariant under Lagrangian isotopy.

A surface T is Hamiltonian stationary for area if

(1)  \frac{d}{dt}|_{t=0} |T_{t}| = ∫_{T} div_{T} X = ∫_{T} −⟨H, X⟩ = 0

for any Hamiltonian vector field X = Jdf, f : T → R. Here T_{t} is a Hamiltonian isotopy starting from T_{0} = T with initial velocity X, and div_{T} X is the divergence of X along T. It follows by rotating into T_{p}T and integrating by parts that

d^{∗}h = 0,

on a Hamiltonian stationary surface.

Define the action 1-form by

α̃ := \frac{1}{2} ∑_{i=1,2}(x^{i}dy^{i} − y^{i}dx^{i}), α := i_{T}^{∗} (α̃)

where i_{T} is the inclusion map of the Lagrangian surface T. Then dα̃ = ω, and dα = 0, so α represents a cohomology class on T. Note that

α̃|_{p} = \frac{Jp}{2}, p ∈ C^{2}, α|_{p} = \frac{Jp^{⊥}}{2} , p ∈ T,

where p^{⊥} is the projection of p to T_{p}T. The 1-form α records the symplectic area enclosed by a given curve γ, since by Stokes’ Theorem

∫_{γ} α = ∫_{D} ω,

where D is any surface in C^{2} with ∂D = γ. It follows that the class [α] is invariant under Hamiltonian variations of T.

Our Clifford torus is the surface

T_{c} := S^{1}(\sqrt{2}) × S^{1}(\sqrt{2}) ⊆ C × C,

which lies in a sphere of radius 2 and has area 8π^{2}. (The usual Clifford torus lies in the unit sphere and has area 2π^{2}.) Write γ_{1}, γ_{2} for curves homotopic to each of the two S^{1} factors. Then

(2)  ∫_{γ_{1}} h = ∫_{γ_{2}} h = ∫_{γ_{1}} α = ∫_{γ_{2}} α = 2π,

so h = α on the Clifford torus T_{c}. For any T ∼ T_{c}, by the invariance of [h] and [α], we still have

(3)  [h] = [α]

in H^{1}(T).

Acknowledgements. We would like to thank W. Minicozzi and P. Topping for helpful discussions.

Proof of the Main Theorem

Let g be the metric induced on T by the inclusion in C^{2}, and let g_{0} be the conformally equivalent flat metric of the same area. Quantities associated with g_{0} will be given a subscript.

Lemma. Let T ∼ T_{c} and suppose T is Hamiltonian stationary for area. Then

g_{0} := |h|^{2}g.

Furthermore, h is parallel with respect to g_{0}, and |h|_{0} = 1.

Proof. Recall that stationarity implies

dh = 0, d^{∗}h = 0

in the metric g. These equations are conformally invariant, so they hold equally well in the flat metric g_{0}. Thus h is a harmonic 1-form, and therefore parallel, with respect to the flat metric g_{0}. In particular, |h|_{0} is constant.

Let us deduce this constant following Minicozzi [M]. By (3), h − α is exact on T, so the vector field

X := J(h − α) = H + \frac{p^{⊥}}{2}

is Hamiltonian. We insert X into (1) to obtain Minicozzi’s Lemma, which avers

∫_{T} |H|^{2} = ∫_{T} − ⟨H, \frac{p^{⊥}}{2} ⟩ = ∫_{T} div_{T} (\frac{p}{2}) = |T |,

for a Hamiltonian stationary surface in the Hamiltonian isotopy class of T_{c}. Evidently, then

|T |_{0} = |T | = ∫_{T} |h|^{2} dμ = ∫_{T} |h|_{0}^{2} dμ_{0} = |h|_{0}^{2}|T |_{0},

which shows |h|_{0} = 1. This implies that

\frac{g_{0}}{g} = \frac{|h|^{2}}{|h|_{0}^{2}} = |h|^{2}.

Next, we apply the Isoperimetric Inequality to get a lower bound for the length of curves in T. Since α records a minimum area spanned by a given curve, the curves can’t get too short. Let γ_{1} and γ_{2} now denote isotoped versions of the original generating circles.

Lemma (Lower Bound of Curves). Let γ be a curve in the homology class kγ_{1} + ℓγ_{2}, k, ℓ ∈ Z. Then

(4)  |γ|_{0}^{2} ≥ 8π^{2}|k + ℓ|.

Remark. Since inequality (4) is quadratic on one side and linear on the other, the best information is obtained for small values of k and ℓ. Of course, (4) can’t see curves in the kernel of [α], but we will get around this later.

Proof. Let D be an area-minimizing disk in C^{2} spanned by γ [ref?]. By the Isoperimetric Inequality [ref?] and (2), we have

|γ|^{2} ≥ 4π|D| ≥ 4π |∫_{D} ω| = 4π |∫_{γ} α| = 4π(2π|k + ℓ|) = 8π^{2}|k + ℓ|.

The proof is completed by applying the next Lemma.

Lemma. Suppose that (T, g_{0}) is a flat 2-torus and (T, g) a conformally equivalent torus of the same area. Then in each homotopy class β, we have

inf_{γ∈β} |γ| ≤ inf_{γ∈β} |γ|_{0}.

Proof. This Lemma should be well known. [ref?] There are two convenient proofs.

(1) Under the Ricci flow

\frac{∂g}{∂t} = (−R + \frac{1}{|T |} ∫_{T} R dμ_{t})

the area is preserved and g converges smoothly to g_{0} as t → ∞ [Ha1]. Furthermore, the length of the shortest geodesic in each homotopy class is nondecreasing [Ha2], so we obtain the Lemma.

(2) Alternately, we can prove it directly by Fubini’s theorem and the Schwarz inequality. Let γ_{0} be a shortest g_{0}-geodesic in the class β, with length a := |γ_{0}|_{0}, and let let γ_{t}, 0 ≤ t ≤ b be parallel copies of γ_{0} that foliate P. We can choose the parameter t so that dμ_{0} = ds dt where ds is g_{0}-arclength along γ_{t}. Then |T |_{0} = ab, and writing g = λ^{2}g_{0}, we have

∫_{0}^{b} |γ_{t}| dt = ∫_{0}^{b} ∫_{0}^{a} λ ds dt
≤ (∫_{0}^{b} ∫_{0}^{a} λ^{2} ds dt)^{1/2} (∫_{0}^{b} ∫_{0}^{a} ds dt)^{1/2}
= |T |^{1/2}|T |_{0}^{1/2} = |T |_{0} = ab,

so

inf_{γ∈β} |γ| ≤ \frac{1}{b} ∫_{0}^{b} |γ_{t}| dt = \frac{ab}{b} = |γ_{0}|_{0},

which proves the Lemma.

Proof of Main Theorem. Let T ∼ T_{c} be Hamiltonian stationary for area. We must prove |T | ≥ 8π^{2}.

Let T̃ be the universal cover of T, equipped with the lift of the flat metric g_{0} by the covering map π. Pick an origin 0 in T̃ and let Γ be the lattice π^{−1}(π(0)). Then T̃ gets a vector space structure and T = T̃/Γ.

Let γ_{1}, γ_{2} be the generators of Γ isotopic to the original generators, so that a typical element of Γ has the form

γ = kγ_{1} + ℓγ_{2}

using vector space addition in T̃.

Let θ: T̃ → R be the potential with θ(0) = 0, dθ = h = α. Let L_{a} be the straight line {θ = a} in T̃. Since |dθ|_{0} = 1, the lines

(5)  …, L_{0}, L_{2π}, L_{4π}, …

ara parallel lines at distance 2π. According to (2),

θ(kγ_{1} + ℓγ_{2}) = ∫_{0}^{kγ_{1}+ℓγ_{2}} h = 2π(k + l),

so Γ ∩ L_{2π} consists of the points

…, γ_{1} − δ, γ_{1}, γ_{1} + δ, γ_{1} + 2δ, …

with δ := γ_{2} − γ_{1}, and constant spacing equal to |δ_{|}0.

Let δ_{1}, δ_{2} be the orthonormal basis of T̃ with δ_{1} := δ/|δ_{0} ∈ L_{0}, ⟨δ_{2}, ·⟩_{0} = h.

Fundamental Domain.

By (4), each point γ_{m} := γ_{1} + (m − 1)δ in Γ ∩ L_{2π} has length at least

|γ_{m}|_{0} ≥ 2\sqrt{2}π,

whereas

⟨γ_{m}, δ_{2}⟩ = 2π,

so by the Pythagorean theorem, |⟨γ_{m}, δ_{1}⟩_{0}| ≥ 2π. Then there exists a unique m such that

⟨γ_{m}, δ_{1}⟩_{0} ≤ −2π, ⟨γ_{m+1}, δ_{1}⟩_{0} ≥ 2π,

which implies

|δ|_{0} = |γ_{m+1} − γ_{m}|_{0} ≥ 4π,

and by (5),

dist_{0}(γ_{m+1} + γ_{m}, L_{0}) = 4π,

so using the parallelogram with corners 0, γ_{m}, γ_{m+1}, γ_{m} +γ_{m+1} as a fundamental domain, we find that

|T | = |T |_{0} ≥ \frac{1}{2}(4π)(4π) = 8π^{2},

as required.

Questions. 1. Is the Clifford torus the unique Hamiltonian stationary torus in its Hamiltonian isotopy class?

2. Is there a Hamiltonian isotopy of C^{n} that takes T_{c} to T_{c}, while taking γ_{m} to γ_{m+k}, for some k ≠ 0?

If not, then the value of m in the proof gives an integer invariant on the class of Hamiltonian stationary tori Hamiltonian isotopic to the Clifford torus. If so, then it is likely that there is a saddle point somewhere along an optimized isotopy. Of course, both alternatives will be moot if the Clifford torus is unique.

References

[Ha1] Hamilton, R., “The Ricci Flow on Surfaces,” Contemp. Math. 71 (1988) 237-261.

[Ha2] Hamilton, “An Isoperimetric Estimate for the Ricci Flow on the Two-Sphere,” Collection: Modern Methods in Complex Analysis (Princeton, NJ, 1992), Annals of Math. Stud. #137, Princ. Univ. Press, Princeton, NJ, 1995, 191-200.

[Mi] Minicozzi, W. P. II, “The Willmore Functional on Lagrangian Tori: Its Relation to Area and Existence of Smooth Minimizers,” J. Amer. Math. Soc. 8 (1995) 761-791.

[Mo] Morvan, J.-M., “Minimal Lagrangian Submanifolds: A Survey,” Collection: Geometry and Topology of Submanifolds, III (Leeds, 1990), World Sci. Pub., L. Verstraelen and A. West, ed., River Edge, NJ, 1991, 206–226.

[O1] Oh, Yong-Geun, “Second Variation and Stabilities of Minimal Lagrangian Submanifolds in Kähler Manifolds,” Invent. Math. 101 (1990) 501–519.

[O2] Oh, Y.-G., “Volume Minimization of Lagrangian Submanifolds under Hamiltonian Deformations,” Math. Zeit. 212 (1993) 175–192.

[SW] Schoen, R., and Wolfson, J., “Minimizing Volume among Lagrangian Submanifolds,”

preprint, 1998.

[T] Topping, P., “Towards the Willmore Conjecture,” preprint, June 1998.

[W] Willmore, T. J., “A Survey on Willmore Immersions,” Geometry and Topology of Submanifolds, IV (Leuven, 1991), F. Dillen and L. Verstraelen, ed., World Sci. Pub., River Edge, NJ, 1992, 11–16.