Unknottedness of real Lagrangian tori in $S^{2}\times S^{2}$

An even dimensional smooth manifold $M$ equipped with a closed non-degenerate 2-form $\omega$ is a symplectic manifold. By Darboux’s theorem [34, Theorem 3.2.2], symplectic manifolds are locally standard, and hence only global properties in symplectic topology are interesting; in particular, the study of middle dimensional submanifolds along which the symplectic form vanishes, namely Lagrangian submanifolds.

In 1986, as one of the first steps in symplectic topology [1, Section 6], Arnold proposed the Lagrangian knot problem asking whether two given Lagrangians are isotopic. As formulated in a survey of Eliashberg–Polterovich [22], there are different flavors of isotopy, namely smooth, Lagrangian and Hamiltonian. Hamiltonian isotopies are Lagrangian, and Lagrangian isotopies are smooth. Two Lagrangians are said to be unknotted if they are isotopic to each other in one of these three ways.

A remarkable result of Gromov [25] says that there are no closed exact Lagrangians in $({\mathbb{R}}^{2n},\sum_{i=1}^{n}dx_{i}\wedge dy_{i})$, and hence the extensive study of Lagrangian tori in ${\mathbb{R}}^{2n}$ has been made for a long time. Chekanov [9] first constructed a monotone Lagrangian torus in ${\mathbb{R}}^{2n}$ for $n\geq 2$, which is Lagrangian isotopic while not Hamiltonian isotopic to the Clifford torus $\mathbb{T}^{n}_{\operatorname{Clif}}=\times_{n}S^{1}$, i.e., products of circles in ${\mathbb{R}}^{2}$ of equal radius. This exhibits that a Lagrangian isotopy cannot be in general deformed into a Hamiltonian isotopy. His result is even more interesting since the classical invariants cannot detect this phenomenon. Indeed, the Audin conjecture [2, Section 6.4], which is proved by Polterovich [37] and Viterbo [40] in dimension 4 and Cieliebak–Mohnke [11] in any dimension, says that the minimal Maslov number (one of the classical invariants) of any Lagrangian torus in ${\mathbb{R}}^{2n}$ is always two, so exotic monotone Lagrangian tori are hard to discover. Auroux [3] constructed infinitely many monotone Lagrangian tori in ${\mathbb{R}}^{6}$ up to Hamiltonian isotopy, while all of them are Lagrangian isotopic. We refer to the work of Dimitroglou Rizell–Evans [17, Corollary C] about the smooth unknottedness of monotone Lagrangian tori inside ${\mathbb{R}}^{2n}$ for $n\geq 5$ odd.

Since Lagrangian 2-planes in ${\mathbb{R}}^{4}$ that are asymptotically linear are trivial by Eliashberg–Polterovich [21] (see also [20]), one may expect reasonable unknottedness results in symplectic 4-manifolds. Symplectic Field Theory developed by Eliashberg–Givental–Hofer [19] has become a core technique to address the Lagrangian unknottedness problems. By a neck-stretching argument [5], Hind [26] showed the Hamiltonian unknottedness of spheres in $S^{2}\times S^{2}$.

In contrast to spheres, the Hamiltonian unknottedness of monotone Lagrangian tori in $S^{2}\times S^{2}$ fails. Chekanov–Schlenk [8] found a monotone Lagrangian torus in $S^{2}\times S^{2}$ which is not Hamiltonian isotopic to the Clifford torus $\mathbb{T}_{\operatorname{Clif}}=S^{1}\times S^{1}$, defined as the product of the equators. This torus, in another description, was also found by Entov–Polterovich [23, Example 1.22]. Vianna [39] showed that there are infinitely many Hamiltonian isotopy classes of monotone Lagrangian tori in $S^{2}\times S^{2}$. Dimitroglou Rizell–Goodman–Ivrii [18, Theorem A] established the Lagrangian unknottedness of tori in $S^{2}\times S^{2}$, namely Lagrangian tori in $S^{2}\times S^{2}$ are unique up to Lagrangian isotopy. As a result, infinitely many monotone Lagrangian tori in $S^{2}\times S^{2}$ are Hamiltonianly knotted.

In this paper, we are interested in a class more rigid than monotone Lagrangian tori, namely real Lagrangian tori. By a real Lagrangian submanifold $L$ in a symplectic manifold $(M,\omega)$ we mean a Lagrangian submanifold that is the fixed point set of an antisymplectic involution $R$ of $M$, i.e., $R^{2}=\operatorname{id}_{M}$ and $R^{*}\omega=-\omega$. The fixed point set $\operatorname{Fix}(R)=\{x\in M\mid R(x)=x\}$ of an antisymplectic involution $R$ of a symplectic manifold $(M,\omega)$ is Lagrangian if it is nonempty. If $(M,\omega)$ is monotone, then every real Lagrangian of $M$ is monotone, see Lemma 2.1. Recall that the Clifford torus $\mathbb{T}_{\operatorname{Clif}}=S^{1}\times S^{1}$ is a real Lagrangian torus in $S^{2}\times S^{2}$ whose antisymplectic involution $R_{\operatorname{Clif}}$ is given by the product of the reflection of $S^{2}$ fixing the equator.

The main result of this paper is to prove the Hamiltonian unknottedness of real Lagrangian tori in $S^{2}\times S^{2}$ in contrast to the monotone case.

As we discussed, being real plays a key role, and the result shows a non-trivial phenomenon of Hamiltonian unknottedness. An immediate consequence of the main theorem together with a known result [30, Proposition B] is the complete classification of real Lagrangian submanifolds in $S^{2}\times S^{2}$.

From a real symplectic perspective, the study of real Lagrangian tori in $S^{2}\times S^{2}$ is the simplest non-trivial case in dimension 4. For topological reasons, there is no closed real Lagrangian in ${\mathbb{R}}^{2n}$ at all. Known monotone symplectic 4-manifolds containing real Lagrangian tori are $S^{2}\times S^{2}$ and the three-fold monotone blow-up of ${\mathbb{C}}P^{2}$, see [7, Theorem D]. In [29], it was proved that the Chekanov–Schlenk torus in $S^{2}\times S^{2}$ is not real, from which we believed that our main result holds. We refer to a recent work of Brendel [6] that extends the result in [29].

In order to prove the main theorem, we employ Gromov’s foliation theorem (Section 2.2) together with the Cieliebak–Schwingenheuer criterion (Section 2.4). Gromov’s strategy was to foliate the monotone symplectic manifold $S^{2}\times S^{2}$ by a family of $J$-holomorphic spheres of minimal symplectic area, and hence to reduce symplectic questions to two dimensional fibered versions. In particular, when one deforms the split complex structure $i\oplus i$ on $S^{2}\times S^{2}$ to any tame almost complex structure $J$, two transversal foliations by $J$-holomorphic spheres still exist (even smoothly depending on $J$), and yield a symplectic $S^{2}$-fibration of $S^{2}\times S^{2}$ over $S^{2}$. In the case of studying a monotone Lagrangian torus, the nicest situation is when a fiber symplectic sphere intersects the torus along a circle or does not intersect at all. Cieliebak–Schwingenheuer provided a criterion for the Hamiltonian unknottedness of tori in $S^{2}\times S^{2}$ by means of this fibered structure. In particular, the existence of two suitable symplectic sections of the fibration is the precise condition for a given monotone Lagrangian torus being Hamiltonianly deformable into the Clifford torus $\mathbb{T}_{\operatorname{Clif}}$. In general, one symplectic section always exists (essentially by an SFT argument, see Proposition 2.6), but the second symplectic section may be missing. For real Lagrangian tori, an antisymplectic involution will provide the second symplectic section as desired. To realize this heuristic argument precisely, we should deal with an almost complex structure that is compatible with an antisymplectic involution.

It might be interesting to mention interactions between real symplectic topology and real algebraic varieties. We refer to [14, 16, 28] for expositions about the topology of real algebraic varieties. Inspired by the notion of quasi-simplicity for real algebraic varieties [28], we formulate a counterpart in symplectic topology. A closed monotone symplectic manifold $(M,\omega)$ is called symplectically quasi-simple if the diffeomorphism type of connected real Lagrangian submanifolds of $M$ uniquely determines its Hamiltonian isotopy class.

Notice that one has to impose a condition on the homology classes of real Lagrangians if necessary. It is known that the problem is true for ${\mathbb{C}}P^{2}$ [30, Proposition A] and $S^{2}\times S^{2}$ (Theorem A). Note that quasi-simplicity in real algebraic varieties is known for all real del Pezzo surfaces by Degtyarev–Itenberg–Kharlamov [14, 15]. Together with Brendel and Moon [7, Theorem D], we obtain the complete list of diffeomorphism types of connected real Lagrangians in toric symplectic del Pezzo surfaces. For symplectic results, we refer to the works of Evans [24] and Seidel [38], which deal with the (un)knotting problems of Lagrangian spheres in del Pezzo surfaces. See also the work of Borman–Li–Wu [4]. Finally, we mention a work of Welschinger [41] which might be related to the problem. In that paper, he defines an invariant under deformation of real symplectic 4-manifolds, called Welschinger invariant.

This paper is organized as follows. In Section 2, we give results relevant for the proof of the main theorem. In Section 3, we prove the main theorem.

Recall that a symplectic manifold $(M,\omega)$ is monotone if there exists $C>0$ such that $c_{1}(M)=C\cdot[\omega]$ and that a Lagrangian submanifold $L$ in $M$ is monotone if there exists $C^{\prime}>0$ such that $\mu_{L}(\beta)=C^{\prime}\cdot\omega(\beta)$ for all $\beta\in\pi_{2}(M,L)$, where $\mu_{L}\colon\pi_{2}(M,L)\to{\mathbb{Z}}$ denotes the Maslov class of $L$, see [34, Definition 3.4.4].

Throughout this paper, the symplectic manifold $S^{2}\times S^{2}$ is equipped with the split symplectic form $\omega\oplus\omega$, where $\omega$ is a Euclidean area form on $S^{2}$. This symplectic form is monotone, and hence every real Lagrangian in $S^{2}\times S^{2}$ is monotone.

Throughout this section, we let $L=\operatorname{Fix}(R)$ be a real Lagrangian torus in $S^{2}\times S^{2}$ for some antisymplectic involution $R$ of $S^{2}\times S^{2}$. Recall that every $J$-holomorphic sphere of $S^{2}\times S^{2}$ in the homology class $A_{i}$ for $i=1,2$ is embedded, see Remark 2.5. We start with the following simple observation.

An immediate consequence of Lemma 3.1 together with Proposition 2.6 is that the real Lagrangian torus $L=\operatorname{Fix}(R)$ is fibered by Gromov’s foliations $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ associated to some $J\in\mathcal{J}_{R}$. More precisely, we have the following.