Constructing exact Lagrangian immersions with few double points

In this paper we study the problem of constructing Lagrangian immersions with the minimal possible number of transverse self-intersection points. It is well known that the existence of a Lagrangian embedding imposes strong topological constraints (e.g. Gromov’s theorem about non-existence of exact Lagrangian submanifolds in standard symplectic $2n$-space, $\mathbb{R}^{2n}_{\mathrm{st}}=(\mathbb{R}^{2n},\sum_{i=1}^{n}dx_{i}\wedge dy_{i})$), and also that, in many cases, two Lagrangian submanifolds must intersect in more points than is suggested by topological intersection theory alone (e.g. results confirming Arnold’s conjectures). In view of this, it was expected that there should be similar lower bounds on the minimal number of double points for Lagrangian immersions, e.g. that an exact Lagrangian immersion of an $n$-torus into $\mathbb{R}^{2n}_{\mathrm{st}}$ with transverse self-intersections would have at least $2^{n-1}$ double points.

Bounds of this kind have been proved for Lagrangian immersions satisfying additional conditions. For instance, it was shown in [6, 8] that any self-transverse exact Lagrangian immersion $f\colon L\to\mathbb{R}^{2n}_{\mathrm{st}}$ of a closed $n$-manifold, for which the Legendrian lift $\tilde{f}\colon L\to\mathbb{R}^{2n}_{\mathrm{st}}\times\mathbb{R}$ has Legendrian homology algebra that admits an augmentation, satisfies the following analogue of the Morse inequalities: the number of double points of $f$ is at least $\frac{1}{2}\sum_{j=0}^{n}\mathrm{rank}(H_{j}(L))$.

Here we prove a surprising result in sharp contrast to such lower bounds: if no extra constraints are imposed then Lagrangian immersions into symplectic manifolds of dimension $>4$ with nearly the minimal number of self-intersection points satisfy a certain $h$-principle. Let us introduce the notation needed to state the result. Let $X$ be an oriented $2n$-manifold and $f\colon L\to X$ a proper smooth immersion of a connected $n$-manifold. If $L$ is non-compact we will assume that $f$ is an embedding outside of a compact set. Following Whitney [22], if $f$ is self-transverse we assign a self-intersection number $I({f})$ to $f$, where $I({f})$ is the mod 2 number of self-intersection points if $n$ is odd or $L$ is non-orientable, and the algebraic number of self-intersection points counted with their intersection signs if $n$ is even and $L$ is orientable. Note that $I(f)$ depends only on the regular homotopy class of $f$ provided that $L$ is closed or that the regular homotopy is an isotopy at infinity. When $n>2$ and $X$ is simply connected, a theorem of Whitney [22] asserts that an immersion $f$ is regularly homotopic to an embedding if and only if $I(f)=0$. If $f\colon L\to X$ is a self-transverse immersion, then we let $\mathrm{SI}(f)\geq 0$ denote its total number of double points. Clearly, $\mathrm{SI}(f)\geq|I(f)|$.

Similarly, given an orientable $(2n-1)$-dimensional manifold $Y$ and an $(n-1)$-dimensional manifold $\Lambda$, consider a smooth regular homotopy $h_{t}\colon\Lambda\to Y$, $0\leq t\leq 1$, which connects embeddings $h_{0}$ and $h_{1}$ and is an isotopy outside of a compact set. Let $H\colon\Lambda\times[0,1]\to Y\times[0,1]$ be given by $H(x,t)=(h_{t}(x),t)$; then $H$ is an immersion. We say that the regular homotopy $h_{t}$ has transverse intersections if the immersion $H$ has transverse double points. In this case we define $I(\{h_{t}\}_{t\in[0,1]}):=I(H)$ and $\mathrm{SI}(\{h_{t}\}_{t\in[0,1]}):=\mathrm{SI}(H)$. Note that if $n$ is even and $\Lambda$ is orientable then $I(\{h_{t}\}_{t\in[0,1]})=-I(\{h_{1-t}\}_{t\in[0,1]})$.

Consider next a Lagrangian regular homotopy, $f_{t}\colon L\to X$, $0\leq t\leq 1$, and write $F\colon L\times[0,1]\to X$ for $F(x,t)=f_{t}(x)$. Let $\alpha$ denote the 1-form on $L\times[0,1]$ defined by the equation $\alpha:=\iota_{\partial/\partial t}(F^{*}\omega)$, where $\iota$ denotes contraction and $t$ is the coordinate on the second factor of $L\times[0,1]$. Then the restrictions $\alpha_{t}:=\alpha|_{L\times\{t\}}$ are closed for all $t$. We call the Lagrangian regular homotopy $f_{t}$ a Hamiltonian regular homotopy if the cohomology class $[\alpha_{t}]\in H^{1}(L)$ is independent of $t$. The following theorem is our main result:

Theorem 1.1 is proved in Section 4.1 as an application of results in [13].

This result has the following consequences for exact Lagrangian immersions into $\mathbb{R}^{2n}_{\mathrm{st}}$. Let $L$ be an $n$-dimensional closed manifold. Recall that according to Gromov’s $h$-principle for Lagrangian immersions the triviality of the complexified tangent bundle $TL\otimes\mathbb{C}$ is a necessary and sufficient condition for the existence of an exact Lagrangian immersion $L\to\mathbb{R}^{2n}_{\mathrm{st}}$, while exact Lagrangian regular homotopy classes are in natural one to one correspondence with homotopy classes of trivializations of $TL\otimes\mathbb{C}$. We write $s(L)$ for the minimal number of double points of a self-transverse exact Lagrangian immersion $L\to\mathbb{R}^{2n}_{\mathrm{st}}$. Given a homotopy class $\sigma$ of trivializations of $TL\otimes\mathbb{C}$, the refined invariant $s(L,\sigma)$ denotes the minimal number of double points of an exact self-transverse Lagrangian immersion $L\to\mathbb{R}^{2n}_{\mathrm{st}}$ representing the exact Lagrangian regular homotopy class $\sigma$.

Corollary 1.3 is proved in Section 4.2; in Section 4.3 we then give more detailed information about the case when $n$ is odd, respectively discuss the Lagrangian embeddings obtained from these immersions by Lagrange surgery. The case $n=2$ in both (1) and (3) above does not follow from our proof of Theorem 1.1; rather, these are results of Sauvaget [20], who gave direct geometric constructions of self-transverse exact Lagrangian immersions of both oriented and non-oriented surfaces. In particular, Sauvaget constructed, as the key point for his result, an exact immersed genus two surface in $\mathbb{C}^{2}$ with exactly one double point. In Appendix A we describe a higher dimensional analogue of that construction.

It is interesting to compare Corollary 1.3 with the results of [9, 10] which show that any homotopy $n$-sphere $\Sigma$ that admits a Lagrangian immersion into $\mathbb{R}^{2n}_{\mathrm{st}}$ with exactly one transverse double point of even Maslov grading and with induced trivialization of the complexified tangent bundle homotopic to that of the Whitney immersion of the standard $n$-sphere¹¹1i.e. the trivializations $\sigma\colon TS^{n}\otimes\mathbb{C}\to\mathbb{C}^{n}$ and $\widetilde{\sigma}\colon T\Sigma\otimes\mathbb{C}\to\mathbb{C}^{n}$ are related as $\widetilde{\sigma}=\sigma\circ(h\otimes\mathbb{C})$, where $h\colon T\Sigma\to TS$ is a bundle isomorphism covering a homotopy equivalence $\Sigma\to S^{n}$., must bound a parallelizable $(n+1)$-manifold. If $n$ is even, both the Maslov grading condition and the homotopy condition are automatically satisfied, and moreover the standard $n$-sphere is the only homotopy $n$-sphere that bounds a parallelizable $(n+1)$-manifold. Thus, if $n$ is even the standard $n$-sphere is the only homotopy $n$-sphere that admits a self-transverse Lagrangian immersion into Euclidean space with only one double point. This means in particular that in the case when $\dim(L)$ is even and $\chi(L)>0$, $s(L)$ is generally not determined by the homotopy type of $L$. The following result constrains the homotopy type of a manifold for which this phenomenon may occur.

Theorem 1.4 is proved in Section 4.4. The proof uses lifted linearized Legendrian homology, introduced in [10] following ideas in [5]. Note that this result can be viewed as an obstruction to an $h$-principle for exact Lagrangian immersions having the minimal, rather than near-minimal, number of self-intersection points.

In [19] Polterovich describes a local Lagrangian surgery construction which resolves a double point of a Lagrangian immersion. Let $Q_{+}$ denote the manifold $S^{1}\times S^{n-1}$, and $Q_{-}$ the mapping torus of an orientation-reversing involution of $S^{n-1}$. Given a Lagrangian immersion $f\colon L\rightarrow X$ of an oriented $n$-manifold with a single double point $p$, [19, Propositions 1 & 2] imply:

1. (1)

   if $n$ is odd, there are Lagrangian embeddings $L\#Q_{\pm}\rightarrow X$;

2. (2)

   if $n$ is even, there is a Lagrangian embedding $L\#Q_{\epsilon}\rightarrow X$, where the sign of $\epsilon$ is given by $(-1)^{n(n-1)/2+1}\textrm{sign}(p)$,

where sign$(p)\in\{\pm 1\}$ denotes the intersection index of the double point. Combining this with Corollary 1.3 yields:

The question of determining the minimal number $k$ for which there is a Lagrangian embedding $Y\#k(S^{1}\times S^{2})\rightarrow\mathbb{R}^{6}_{\mathrm{st}}$ was raised explicitly by Polterovich [19, Remark 2]. The construction in Appendix A, in combination with Lagrange surgery, gives an explicit Lagrangian embedding of $Q_{+}\#Q_{+}\#Q_{+}$ into $\mathbb{R}^{2n}_{\mathrm{st}}$ for any odd $n\geq 3$.

The (non-)existence question for Lagrangian embeddings into $\mathbb{R}^{2n}_{\mathrm{st}}$ with vanishing Maslov class is a well-known problem in symplectic topology. Viterbo proved in [21] that if $L$ admits a metric of non-positive sectional curvature then any Lagrangian embedding $L\rightarrow\mathbb{R}^{2n}_{\mathrm{st}}$ has non-trivial Maslov class, whilst Fukaya, Oh, Ohta and Ono infer the same conclusion in Theorem $K$ of [15] whenever $L$ is a spin manifold with $H^{2}(L;\mathbb{Q})=0$. Corollary 1.6 shows that the hypotheses in these theorems play more than a technical role; in particular, the assumption on second cohomology in the latter result cannot be removed. Note that by taking products of the Maslov zero $S^{1}\times S^{2}$ in $\mathbb{R}^{6}_{\mathrm{st}}$, one obtains Maslov zero Lagrangian embeddings in $\mathbb{R}^{6k}_{\mathrm{st}}$ for every $k>1$. We point out that the Lagrangian embedding in Corollary 1.6 is not monotone. In the monotone case, the Maslov index of the generator of the first homology of positive action necessarily equals $2$, see [14, Proposition 2.10] (and references therein) and [5, Theorem 1.5 (b)].

The paper is organized as follows. In Section 2 we recall the notions and the main results of the theory of loose Legendrian knots from [18]. In Section 3 we formulate the $h$-principle for Lagrangian embeddings with conical singularities, established in [13], and deduce from it its own generalization: in the presence of a conical singularity, Lagrangian immersions with the minimal number of self-intersections abide an $h$-principle. Theorem 1.1 is then proved in Section 4.1 as an application of this $h$-principle. Corollary 1.3 and related explicit results about Lagrangian immersions into $\mathbb{R}^{2n}_{\mathrm{st}}$ are proved in Sections 4.2, 4.3, and 4.4. As is typical with $h$-principles, Theorem 1.1 does not provide explicit constructions of Lagrangian immersions with the minimal number of double points. In Appendix A we complement Theorem 1.1 by constructing an explicit exact Lagrangian immersion of $P=(S^{1}\times S^{n-1})\#(S^{1}\times S^{n-1})$ into $\mathbb{R}^{2n}_{\mathrm{st}}$ with exactly one transverse double point (in particular, yielding an immersion in each dimension violating the Arnold-type bound which pertains in the presence of linearizable Legendrian homology algebra). Our construction is a generalization of Sauvaget’s construction in the case $n=2$. The construction of the appendix seems to have no elementary relation to the arguments earlier in the paper; in general, our immersions with few double points are obtained by first introducing many double points and then canceling them in pairs, whilst in the Appendix considerable effort is expended to keep the Lagrangians swept by appropriate Legendrian isotopies embedded.

Acknowledgements. T.E. and I.S. are grateful to François Laudenbach for drawing their attention to Sauvaget’s work. Y.E. is grateful to the Simons Center for Geometry and Physics where a part of this paper was completed. The authors thank Alexandr Zamorzaev for pointing out a gap in the original proof of Lemma 3.4.

We start with a discussion of the stabilization construction for Legendrian submanifolds, see [11, 4, 18].

Consider standard contact $\mathbb{R}^{2n-1}$:

where $(x_{1},y_{1},\dots,x_{n-1},y_{n-1},z)$ are coordinates in $\mathbb{R}^{2n-1}$, with the Legendrian coordinate subspace $\Lambda_{0}\subset\mathbb{R}^{2n-1}_{\mathrm{st}}$ given by

Then $(\mathbb{R}^{2n-1}_{\mathrm{st}},\Lambda_{0})$ is a local model for any Legendrian submanifold in a contact manifold. More precisely, if $\Lambda\subset Y$ is any Legendrian $(n-1)$-submanifold of a contact $(2n-1)$-manifold $Y$ then any point $p\in\Lambda$ has a neighborhood $\Omega\subset Y$ that admits a map

which is a contactomorphism onto a neighborhood of the origin.

We will carry out the stabilization construction in a local model that is slightly different from $(\mathbb{R}^{2n-1}_{\mathrm{st}},\Lambda_{0})$, which we discuss next. Let $F\colon\mathbb{R}^{2n-1}_{\mathrm{st}}\to\mathbb{R}^{2n-1}_{\mathrm{st}}$ denote the contactomorphism,

Then $F$ maps $\Lambda_{0}$ to

In the language of Appendix A.2, the front $\Gamma_{\mathrm{cu}}$ of $\Lambda_{\mathrm{cu}}$ in $\mathbb{R}^{n-1}\times\mathbb{R}$ is the product of $\mathbb{R}^{n-2}\subset\mathbb{R}^{n-1}$ and the standard cusp $\{(x_{1},z)\colon 9z^{2}=8x_{1}^{3}\}$ in $\mathbb{R}\times\mathbb{R}$. In particular, the two branches of the front are graphs of the functions $\pm h$, where

defined on the half-space $\mathbb{R}^{n-1}_{+}:=\{x=(x_{1},\dots,x_{n-1})\colon x_{1}\geq 0\}$.

Let $U$ be a domain with smooth boundary contained in the interior of $\mathbb{R}^{n-1}_{+}$, $U\subset\operatorname{int}(\mathbb{R}^{n-1}_{+})$. Pick a non-negative function $\phi\colon\mathbb{R}^{n+1}_{+}\to\mathbb{R}$ with the following properties: $\phi$ has compact support in $\operatorname{int}(\mathbb{R}^{n-1}_{+})$, the function $\widetilde{\phi}(x):=\phi(x)-2h(x)$ is Morse, $U=\widetilde{\phi}^{-1}([0,\infty))$, and $0$ is a regular value of $\widetilde{\phi}$. Consider the front $\Gamma_{\mathrm{cu}}^{U}$ in $\mathbb{R}^{n-1}\times\mathbb{R}$ obtained from $\Gamma_{\mathrm{cu}}$ by replacing the lower branch of $\Gamma_{\mathrm{cu}}$, i.e. the graph $z=-h(x)$, by the graph $z=\phi(x)-h(x)$. Since $\phi$ has compact support, the front $\Gamma_{\mathrm{cu}}^{U}$ coincides with $\Gamma_{\mathrm{cu}}$ outside a compact set. Consequently, the Legendrian embedding $\Lambda_{\mathrm{cu}}^{U}\colon\mathbb{R}^{n-1}\to\mathbb{R}^{2n-1}$ defined by the front $\Gamma_{\mathrm{cu}}^{U}$ coincides with $\Lambda_{\mathrm{cu}}$ outside a compact set.

To transport the stabilization construction to our standard local model, let $\Lambda_{0}^{U}=F^{-1}(\Lambda_{\mathrm{cu}}^{U})$ and $\Lambda_{0;\,t}=F^{-1}(\Lambda_{\mathrm{cu};\,t})$, where $\Lambda_{\mathrm{cu};\,t}$, $t\in[0,1]$, is the regular Legendrian homotopy constructed in Lemma 2.1. Then $\Lambda_{0;\,t}$, $t\in[0,1]$, is a compactly supported Legendrian regular homotopy connecting $\Lambda_{0}$ to $\Lambda^{U}_{0}$.

Consider a Legendrian $(n-1)$-submanifold $\Lambda$ of a contact $(2n-1)$-manifold $Y$ and a point $p\in\Lambda$. Fix a neighborhood $\Omega\subset Y$ of $p$ and a contactomorphism

Replacing $\Omega\cap\Lambda$ with $\Phi^{-1}(\Lambda_{0}^{U})$ we get a Legendrian embedding $\Lambda^{U}$ which coincides with $\Lambda$ outside of $\Omega$, and replacing it with $\Phi^{-1}(\Lambda_{0;\,t})$ we get a Legendrian regular homotopy $\Lambda_{t}$, $t\in[0,1]$ connecting $\Lambda$ to $\Lambda^{U}$ with $\mathrm{SI}\left(\{\Lambda_{t}\}_{t\in[0,1]}\right)$ equal to the minimal number of critical points of a Morse function on $U$ which attains its minimum value on $\partial U$, and with $I\left(\{\Lambda_{t}\}_{t\in[0,1]}\right)=(-1)^{k-1}\chi(U)$ if $n=2k$ and $I\left(\{\Lambda_{t}\}_{t\in[0,1]}\right)\equiv\chi(U)\;\mathrm{mod}\,2$ if $n$ is odd. We say that $\Lambda^{U}$ is obtained from $U$ via $U$-stabilization in $\Omega$. The most important case for us will be when $U$ is the ball. We say in this case that $\Lambda^{U}$ is the stabilization of $\Lambda$ in $\Omega$ or simply the stabilization of $\Lambda$.

Let $\xi$ denote the contact plane field on $Y$, and note that there is an induced field of (conformally) symplectic 2-forms on $\xi$. We say that two Legendrian embeddings $f_{0},f_{1}\colon\Lambda\to Y$ are formally Legendrian isotopic if there exists a smooth isotopy $f_{t}\colon\Lambda\to Y$ connecting $f_{0}$ to $f_{1}$ and a $2$-parametric family of injective homomorphisms $\Phi_{s,t}:T\Lambda\to TY$ such that $\Phi_{0,t}=df_{t}$ for all $t\in[0,1]$, $\Phi_{s,0}=df_{0}$ and $\Phi_{s,1}=df_{1}$ for all $s\in[0,1]$, and such that $\Phi_{1,t}$ is a Lagrangian homomorphism $T\Lambda\to\xi$ for all $t\in[0,1]$. We will need the following simple lemma, see [11, 4, 18].

Let $Y$ be a contact $(2n-1)$-manifold, $n>2$. We continue using the notation from Section 2.1. A Legendrian embedding $\Lambda\to Y$ of a connected manifold $\Lambda$ (which we sometimes simply call a Legendrian knot) is called loose if it is isotopic to the stabilization of another Legendrian knot. We point out that looseness depends on the ambient manifold. A loose Legendrian embedding $\Lambda$ into a contact manifold $Y$ need not be loose in a smaller neighborhood $Y^{\prime}$, $\Lambda\subset Y^{\prime}\subset Y$.

Any Legendrian submanifold $\Lambda\subset Y$ can be made loose by stabilizing it in arbitrarily small neighborhood of a point. Moreover, it can be made loose even without changing its formal Legendrian isotopy class. Indeed, one can first stabilize it and then $U$-stabilize for some $U$ with $\chi(U)=-1$.

The following result about realizing a Legendrian isotopy as a Lagrangian embedding of a cylinder is proved in [12, Lemma 4.2.5]. Let $Y$ be a contact manifold with contact structure $\xi$ given by the contact $1$-form $\alpha$, $\xi=\mathrm{ker}(\alpha)$. The symplectization of $Y$ is the manifold $\mathbb{R}\times Y$ with symplectic form $d(e^{s}\alpha)$, where $s$ is a coordinate along the $\mathbb{R}$-factor.

We will need the following modification of Lemma 3.1 for Lagrangian immersions. Let $f_{t}\colon\Lambda\to Y$, $t\in[-1,1]$, be a self-transverse regular Legendrian homotopy constant near its endpoints that connects Legendrian embeddings $f_{-}$ and $f_{+}$. As in Lemma 3.1 we extend $f_{t}$ to all $t\in\mathbb{R}$ as independent of $t$ for $|t|\geq 1$.

Let $S^{2n-1}_{\mathrm{st}}=(S^{2n-1},\xi_{\mathrm{st}})$ be the sphere with the standard contact structure $\xi_{\mathrm{st}}$ defined by the restriction $\alpha_{\mathrm{st}}$ to the unit sphere $S^{2n-1}\subset\mathbb{R}^{2n}$ of the Liouville form $\lambda_{\mathrm{st}}:=\frac{1}{2}\sum_{j=1}^{n}(x_{j}dy_{j}-y_{j}dx_{j})$ in $\mathbb{R}^{2n}_{\mathrm{st}}$.

For any integer $m$ we denote by $(r,x)\in(0,\infty)\times S^{m-1}$ polar coordinates in $\mathbb{R}^{m}-\{0\}$, i.e. $x$ is the radial projection of a point to the unit sphere, and $r$ is its distance to the origin. The symplectic form $\omega_{\mathrm{st}}$ in $\mathbb{R}^{2n}$ has the form $d(r^{2}\alpha_{\mathrm{st}})$ in polar coordinates.

A map $h\colon\mathbb{R}^{n}\to\mathbb{R}^{2n}_{\mathrm{st}}$ is called a Lagrangian cone if $h^{-1}(0)=0$ and if it is given by the formula $h(r,x)=(cr^{2},\phi(x))$ in polar coordinates, where $\phi\colon S^{n-1}\to S^{2n-1}_{\mathrm{st}}$ is a Legendrian embedding and $c$ is a positive constant.

Note that there exists a symplectomorphism

given by the formula $C(t,x)=\left(e^{\frac{t}{2}},x\right)$ in polar coordinates. Under this symplectomorphism Lagrangian cones in $\mathbb{R}^{2n}_{\mathrm{st}}$ correspond to cylindrical Lagrangian manifolds in the symplectization $(\mathbb{R}\times S^{2n-1},d(e^{t}\alpha_{\mathrm{st}}))$ of $S^{2n-1}_{\mathrm{st}}$.

Let $L$ be an $n$-dimensional manifold and $X$ a $2n$-dimensional symplectic manifold. A map $f\colon L\to X$ is called a Lagrangian immersion with a conical point at $p\in L$, if $f|_{L-\{p\}}$ is a Lagrangian immersion, and if in a neighborhood of $p$ and in a Darboux chart around $f(p)$, the map is equivalent to a Lagrangian cone $h\colon\mathbb{R}^{n}\to\mathbb{R}^{2n}_{\mathrm{st}}$ near the origin. The Legendrian embedding $\phi\colon S^{n-1}\to S^{2n-1}_{\mathrm{st}}$ corresponding to this cone is called the link of the conical point.

We define a regular Lagrangian homotopy $f_{t}\colon L\to X$, $t\in[0,1]$, of immersions with a conical point at $p$ to be a homotopy which is fixed in some neighborhood of the singular point $p$ and that is an ordinary regular Lagrangian homotopy when restricted to $L-\{p\}$. For a self-transverse immersion $f$ with a conical point at $p$, we define the self-intersection number $I(f)$ as $I(f|_{L\setminus\{p\}})$. Then $I(f)$ is invariant under regular homotopies fixed near $p$.