Akaho, M.

Eliashberg, Y.

Itagaki, Y.

Nishinou, T.

Kyoto University Research Information Repository

TitleTopology of Lagrangian Submanifolds (Proceedings of theWorkshop "Algebraic Geometry and Integrable Systemsrelated to String Theory")Author(s)Eliashberg, Y.; Akaho, M.; Itagaki, Y.; Nishinou, T.Citation数理解析研究所講究録 (2001), 1232: 125-133Issue Date2001-10URL http://hdl.handle.net/2433/41487RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityTopology of Lagrangian SubmanifoldsY. Eliashbergnoted by M. Akaho, Y.Itagaki and T. NishinouY. Eliashberg gave a talk on topology of Lagrangian submanifolds at aconference held at RIMS from 9to 12 May 2000. Here we note only a partof his talk.The content of Sections 1 and 2, except Theorem 1.4 can be found in[1]. Theorem 1.4 is joint with L. Polterovich and is contained in [2]. Resultsstated in Section 3 are extracted from a joint with M. Gromov paper [3].1 Unknotting of Lagrangian surfaces in sym-plectic 4-manif0ldLet $(M^{2n}, \omega)$ be asymplectic manifold. An $\mathrm{n}$-dimensional submanifold $L$ iscalled aLagrangian submanifold if $\omega|_{L}=0$ .Example $M=\mathbb{R}^{2n}=\mathbb{C}^{n}$ , $\omega_{0}=\dot{.}\sum_{=1}^{n}dx^{i}\wedge dy^{i}$ , where $(z_{1}, \cdots, z_{n})=(x_{1}+$$\mathrm{i}\mathrm{y}\mathrm{i}$ , $\cdots$ , $x_{n}+iy_{n})$ is the standard coordinate of $\mathbb{C}^{n}$ , is asymplectic mani-fold. In this case, alinear $\mathrm{n}$-dimensional plane $L$ is Lagrangian if and onlyif $iL[perp] L$ . If instead we have $iL\overline{\mathrm{r}\mathrm{h}}L$ , then $L$ is called totallyreal. Generaltotally real submanifolds are defined in an obvious manner.We will treat $n=2$ case of the above example. The first result we willmention is the following unknottedness theorem:Theorem 1.1. Let $\mathbb{R}_{+}^{4}=\{y_{2}\geq 0\}$ and assume that a 2-disk $\Delta$ is embeddedin $\mathbb{R}_{+}^{4}$ as $(\Delta, \partial\Delta)\subset(\mathbb{R}_{+}^{4}, \partial \mathbb{R}_{+}^{4})$ and $\partial\Delta=\{|z_{1}|=1, z_{2}=0\}$ . Then, if wehave $\omega|_{\Delta}\geq 0$ , then Ais unknotted, $i.e$ . we can isotope $\Delta$ relative to $\partial\Delta$ toa disk in $\partial \mathbb{R}_{+}^{4}$ .The proof of this theorem relies on the method of filling with holomorphicdiscs and we quot the necessary result here. We first define the pseud0-convexity of an oriented hypersurface $\Sigma$ of general symplectic manifold $(M^{2n},\omega$1232 2001 125-133125Let $J$ be an almost complex structure on $M$ tamed by $\omega$ . Then, for everypoint $x$ on $\Sigma$ , the tangent space $T_{x}M$ has a $J$-invariant(2n-2) dimensionalsubspace $T_{x}^{J}\Sigma$ . $\cup T_{x}^{J}\Sigma$ is a(2n- 2) dimensional subbundle $T^{J}M$ of $TM$ .$x\in M$Since $\Sigma$ is oriented and each $T_{x}^{J}\Sigma$ has anatural orientation as acomplexvector space, the quotient 1-dimensional bundle $T\Sigma/T^{J}\Sigma$ is also orientable,i.e. trivial. In particular, there is atrivial sub-line bundle $\underline{\mathbb{R}}$ of $T\Sigma$ such that$T\Sigma=\underline{\mathbb{R}}\oplus T^{J}\Sigma$ . Choosing anon-vanishing section $\eta$ of $\underline{\mathbb{R}}$ fixes a1-form $\alpha$on $\Sigma$ satisfying $\alpha|_{T^{J}\Sigma}=0$ and $\alpha(\eta)>0$ .Definition 1.1. $\Sigma$ is called $J$ -convex, or pseudoconvex if the quadraticform $t\mapsto d\alpha(t, Jt)$ on $T^{J}\Sigma$ is positive definite.With this preparation, we can state the following result.Theorem 1.2. Let 0be a domain in $\mathbb{R}^{4}$ such that an is pseudo convexw.r.L some almost complex structure $J$ tamed by $\omega_{0}$ . Let $F$ be a surfacewith boundar$ry$ embedded in an such that $F$ has a unique complex point whichis elliptic, and $J$ is integrable near that point. Moreover, assume that thereis a $J$ -holomorphic disc Awith $\partial F=\partial\Delta$ and which is transversal to $\partial\Omega$along ab. Then $F\cup\Delta$ can be filled with $a$. family of embedded, disjoint$J$ -holomorphic discs $\{D_{t}\}$ .Now we explain the outline of the proof of the unknottedness theorem. First,we take alarge sphere $S$ in $\mathbb{R}^{4}$ with the center on the $y_{2}$-Bxis which intersectswith the $z_{1}$-plane along ab, and let $B$ be the interior domain of $S$ . We cantake adisk $F$ in $S$ whose boundary coincides with ab and has auniquecomplex point which is elliptic, and moreover it is isotopic to adisk on $\partial \mathbb{R}_{+}^{4}$relative to the boundary. On the otherhand, the disk Acan be slightly de-formed by aboundary fixing isotopy so that $\omega|_{\Delta}>0$ . Taking $B$ large enough,we can suppose that $\Delta$ is contained in $B$ . Then, there is an almost complexstructure $J$ tamed by $\omega_{0}$ for which Ais $J$-holomorphic. Moreover $J$ can bechosen integrable near the elliptic point of $F$ . This will allow us to apply thefilling with holomorphic disc technique to the triple $(\Omega=B, F, \Delta)$ , and thuswill supply us with the isotopy mentioned in the theorem.Using the same technique, we can prove the next theorem.Theorem 1.3. Let $\Pi_{0}$ and $\Pi_{1}$ denote the hyperplanes $\{y_{2}=0\}$ and $\{y_{2}=$$1\}$ , and let $L_{0}$ be the Lagrangian cylinder $\{|z_{1}|=1, x_{2}=0,0\leq y_{2}\leq 1\}$ .Suppose $L$ is another Lagrangian cylinder between $\Pi_{0}$ and $\Pi_{1}$ having thesame boundary as L$. Then, $L$ is Lagrangian isotopic to $L_{0}$ relative to theboundary in $\mathbb{R}^{4}\backslash (D_{+}\cup D_{-}\cup R_{+})$ , where $D_{+}=\{|z_{1}|\leq 1, z_{2}=0\}$ , $D_{-}=$$\{|z_{1}|\leq 1, z_{2}=1\}$ , and $R_{+}=\{y_{2}\geq 1,x_{2}=z_{1}=0\}$ .126(Outline of the proof) We again replace the plane $\Pi_{0}$ by aboundary $\partial\Omega$ of alarge convex domain $\Omega$ such that $\partial\Omega$ intersects with the $z_{1}$-plane along theunit circle $C$ . As before, we can take adisk $F$ whose boundary coincides with$C$ and which has aunique complex point which is elliptic. On the otherhand,we can modify the cylinder $\Delta$ by aboundary fixing isotopy, as well as gluingadisk on the top of it, so that the resulting disk $\Delta$ will $1_{1^{\mathrm{w}}}\mathrm{a}\mathrm{v}\mathrm{e}$ the boundary $C$ ,on which the symplectic form is positive. Then, as before, we can choose analmost complex structure $J$ integrable near the elliptic point of $F$ , tamed by$\omega_{0}$ , with respect to which $\Delta$ is holomorphic, and then apply the filling withholomorphic disks technique to $(F, \Delta)$ . This will supply the isotopy we want.The next is the unknottedness result for Lagrangian knots in $\mathbb{R}^{4}$ .Theorem 1.4. There is no knotted Lagrangian plane in $\mathbb{R}^{4}$ . That is, if $\phi$ :$\mathbb{R}^{2}arrow(\mathbb{R}^{4}, \omega_{0})$ is a Lagrangian embedding which coincides with the inclusion$i$ : $\mathbb{R}^{2}arrow \mathbb{C}^{2}$ defined by $(x, y)\mapsto*(x, 0,0,y)$ outside of a compact set, thenthere is a compact supported Lagrangian isotopy between $\phi$ and $i$ .(outline of the proof) This theorem is aconsequence of the following tworesults.Proposition 1. If a Lagrangian knot $L$ in $\mathbb{R}^{4}$ is contained in some simplehypersurface $Q$ , then $L$ is Lagrangian isotopic to the flat plane.Proposition 2. For every Lagrangian knot $L$ in $\mathbb{R}^{4}$ , there is a simple hy-persurface $Q$ containing itWe first explain the word simple hypersurface. Let $R$ be aoriented hy-persurface in $(\mathbb{R}^{4}, \omega_{0})$ . Then, the symplectic form $\omega_{0}$ restricted to $R$ definesan oriented 1-dimensional distribution on $R$ by $Ker\omega_{0}$ . $R$ integrates into a1-dimensional foliation. We call this foliation characteristic.Definition 1.2. Ahypersurface $Q$ in $\mathbb{R}^{4}$ is called simple if each leaf of itscharacteristic foliation is diffeomorphic to $\mathbb{R}$ and outside acompact set of$Q$ , each leaf coincide with apart of one of parallel straight lines of agivendirection.The proof of proposition 1is carried out by constructing a2-dimensi0naifoliation $\{M_{t}\}_{t\in \mathrm{R}}$ on $Q$ such that each leaf is aLagrangian diffeomorphic to$\mathbb{R}^{2}$ , $M_{0}=L$ and $M_{t}$ are embedded standard $\mathbb{R}^{2}\mathrm{s}$ for $t<-1,t>0$ . It canbe done using the characteristic foliation. As for the proof of proposition2, we need the filling with holomorphic disks technique. Namely, one firsttakes a2-dimensional foliation whose leaves consist of trajectories of the127characteristics foliation which intersect at $-\infty$ aline, parallel to agivendirection. The constructed foliation is not flat at $+\infty$ , but can be flattenvia an appropriate Hamiltonian isotopy. We first fix some notations. Let$(u, v, x, y)$ be the coordinate for $\mathbb{R}^{4}$ , $Q_{0}$ be the hyperplane $\{v=0\}$ , $L_{0}$ bethe standard Lagrangian plane $\{(u, 0,0,y)\}$ and $\Sigma_{0}=L_{0}\cap C$ . Let $C=$$\{(x-u)^{2}+y^{2}\leq 1\}$ and $K=\{(x-u)^{2}+y^{2}\leq 1/2\}$ be two cylinderscontained in $\mathbb{R}^{3}=\{(u, x, y)\}$ . There is aconvex domain $V_{\delta}$ defined by $V_{\delta}=$$\{-\delta\phi(u, x, y)<v<\delta\phi(u, x, y)\}$ where $\delta>0$ and $\phi(u, x, y)=1-(x-y)^{2}-y^{2}$ .It satisfies $\partial V_{\delta}\supset\partial C$ . Then, by asuitable dilatation, we can suppose thatour Lagrangian knot $L$ coincides with $L_{0}$ outside of $K$ and is contained in$V_{\delta}$ . We now isotope $C\cap\{-1\leq u\leq 1\}$ to aset like the figure below.We denote this map by $\Phi$ . This can be done so that the images of the disks$\{t\}\cross D^{2}$ are symplectic. We call the image of the discs by $N$ . Then, thereis asymplectic embedding $\chi$ from aneighbourhood of $N$ to $V$ such that$\chi(\Sigma_{0})=V\cap L$ and $\chi$ is the identity outside $K$ . We can define an almostcomplex structure $J$ on $\mathbb{R}^{4}$ tamed by $\omega_{0}$ such that the image of the disks$\{t\}$ $\cross D^{2}$ by the map $\chi 0\Phi$ are $J$-holomorphic and flat near $\partial$. $V$ and outsideof a compact set in $\mathbb{R}^{4}$ . Then, since $\partial C$ is contained in apseudo convexboundary, examining the Maslov class of the generator of the first homologygroup of $\partial C$ , we see that we can extend $\chi 0\Phi$ to the whole cylinder $C$ inaway that images of the discs $\{t\}\cross D^{2}$ , $t\in \mathbb{R}$ are $J$-holomorphic and for$|t|$ larger than 1, the map on $\{t\}\cross D^{2}$ is the identity. If we call this map$F$ , then $Q=(Q_{0}-C\cap\{-1\leq u\leq 1\})\cup F(\{-1\leq u\leq 1\})$ is the requiredsimple hypersurface1282 Invariants of $S^{2}$-knots in $\mathrm{R}^{4}$ via symplecticgeometryLet $f$ : $S^{2}\mathrm{e}arrow \mathbb{R}^{4}$ be an embedding, and $\alpha:=[f]$ the isotopy class of $f$ . Letus denote by $D(a, b)$ the polydisc $\{(z_{1}, z_{2})\in \mathbb{C}^{2}||z_{1}|\leq a, |z_{2}|\leq b\}$ .We say that the class $\alpha$ admits a $(a, b)$ -realization for $a>1$ , $b>0$if ct can be represented by an embedded sphere $S=\Delta\cup D\subset \mathbb{R}^{4}$ where$D=\{|z_{1}|\leq 1, z_{2}=b\}$ and $\Delta$ is a2-disk satisfying the following properties:$(\Delta, \partial\Delta)\subset(\mathbb{C}^{2}\backslash \mathrm{I}\mathrm{n}\mathrm{t}7)(a, b)$ , $\partial D(a, b))$ intersects $\partial D(a, b)$ transversely alongthe circle $\partial\Delta=\{|z_{1}|=1, z_{2}=b\}$ , and $\omega|_{\Delta}>0$ .$a$Lemma 2.1. For any isotopy class $\alpha$ of embeddings $S^{2}\mathrm{c}arrow \mathbb{R}^{4}$ , there exista $>1$ , b $>0$ such that $\alpha$ admits a(a,$b)-$ realization.The following theorem asserts that asymplectic 2-disc cannot be knottednot only in the half-space but even in the complement of asufficiently largepolydisc.Theorem 2.2. If $[f]$ admits $a(3,2)$ -realization, then it is trivia $l$.129We sketch the proof of this theorem. Set the following notations:$\Omega=\{x_{2}\leq\epsilon|z_{1}|^{2}/(1-\epsilon)^{2}\}$ where $z_{2}=x_{2}+iy_{2}$$\Sigma=\partial\Omega$ $\cap D(a,b)$$A_{c,d}=\{|z_{1}|\leq c, |y_{2}|\leq d\}$$\Sigma_{c,d}=A_{c,d}\cap\Sigma$$G=D(a, b)\backslash (A_{1,\epsilon}\cap\Omega)$$S=\{y_{2}=0, |z_{1}|\leq 1-\epsilon\}\cap \mathrm{C}$ .Deform $\Delta$ into the following form, and denote the resulting disc by $\tilde{\Delta}$ .IThe disc Aintersects Itransversely along $\partial\tilde{\Delta}=\{z_{1}|=1-\epsilon, z_{2}=\epsilon\}$ .We can assume that $\omega|_{\overline{\Delta}}>0$ and Ais holomorphic near ab (with respectto the standard complex structure on $\mathbb{C}^{2}$). Let us choose an almost complexstructure $J$ on $\mathbb{R}^{4}$ such that:$\bullet$ $J$ is tamed by $\omega$ .$\bullet$ $J$ is standard on $G$ , near $\Sigma$ and at infinity.$\bullet$$\Delta-$ is J-holomorphic.Then, the theorem can be deduced from the following:Lemma 2.3. The pair $(S,\tilde{\Delta})$ can be filled with $J$ -holomorphic discs.Let $q\in S$ be the elliptic point of $S$ , and $\{\Delta_{t}\}_{t}$ be aBishop family of$J$-holomorphic disks developing from $q$ . To show the lemma, it is sufficien$\mathrm{t}$130to prove that $\mathrm{I}\mathrm{n}\mathrm{t}\Delta_{t}\cap\Sigma_{\mathrm{I},\epsilon}=\emptyset$ . We want eliminate the following case.$\Sigma_{2,1}$Notice that no disk can be tangent to astrictly pseudoconvex hypersurfacefrom aconvex side.Suppose that some disc $\Delta_{t}$ is tangent to $\Sigma_{2,1}$ at apoint $p$ from the concaveside. Observe that for any $t$ we have$\int_{\Delta_{t}}\omega$ $< \int_{S}\omega=\pi(1-\epsilon)^{2}$ by Stokes’ theorem.On the other hand, holomorphic curves have the following monotonicity prop-erty:Lemma 2.4. Let $C$ be a properly embedded holomorphic curve in the openball $B$ of radius $r$ in $\mathbb{C}^{n}$ . Suppose that $C$ contains the center of B. ThenArea $C\geq\pi r^{2}$ .We apply this lemma to $C=\Delta_{t}$ , $B=B_{1-\epsilon}(p)$ . By assumption, $B\cap\Delta_{t}$is contained in $G$ , and $J$ is standard on $G$ . Therefore$\pi(1-\epsilon)^{2}\leq \mathrm{A}\mathrm{r}\mathrm{e}\mathrm{a}(\Delta_{t}\cap B)\leq\int_{\Delta_{t}}\omega$ .This contradicts the inequality $\int_{\Delta_{t}}\omega$ $<\pi(\dot{1}-\epsilon)^{2}$ .1313Legendrian linking problemLet $V$ be amanifold and $PT^{*}(V)$ the projetivized cotangent bundle, i.e.,the space of all tangent hyperplanes in $T(V)$ . The manifold $PT^{*}(V)$ has acontact structure $\eta\subset T(PT^{*}(V))$ such that lift of each hypersurface $W\subset V$to $PT^{*}(V)$ , denote by $\mathcal{L}_{W}\subset PT^{*}(V)$ , is aLegendrian submanifold for 77.Moreover, let $W\subset V$ be asmooth submanifold of positive codimension. Put$\mathcal{L}_{W}:=\{(w, H_{w})\in PT^{*}(V)|T_{w}(W)\subset H_{w}\subset T_{w}(V)H_{w}\mathrm{i}\mathrm{s}\mathrm{a}\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h}$that$\}$ .Then $\mathcal{L}_{W}$ is also aLegendrian submanifold for 77. Let $W_{1}$ and $W_{2}$ be subman-ifolds properly immersed into $V$ such that they intersect transversely. Here“properly” means “being closed as asubset in $V$” Then $\mathcal{L}_{W_{1}}\cap \mathcal{L}_{W_{2}}=\emptyset$ .Let $\mathcal{L}_{1}(t)$ and $\mathcal{L}_{2}(t)$ be compact supported contact isotopies of $\mathcal{L}_{W_{1}}$ and $\mathcal{L}_{W_{2}}$such that $\mathcal{L}_{1}(1)$ and $\mathcal{L}_{2}(1)$ have disjoint projections to $V$ . We denote by$\#(\mathcal{L}_{1}(t)* \mathcal{L}_{2}(t))$ the minimal number of crossings between all (compact sup-regported) contact isotopies $\mathcal{L}_{1}(t)$ and $\mathcal{L}_{2}(t)$ which intersect transeversely andmove $\mathcal{L}_{1}(0)$ and $\mathcal{L}_{2}(0)$ to $\mathcal{L}_{1}(1)$ and $\mathcal{L}_{2}(1)$ .Theorem 3.1. Suppose $W_{1}\cap W_{2}$ is compact, then we have$\#(\mathcal{L}_{1}(t)\mathrm{X}\mathcal{L}_{2}(t))reg\geq\frac{1}{2}$ rank $H_{*}(W_{1}\mathrm{R} W_{2})$ ,where $W_{1}\mathrm{R}$ $W_{2}$ denote the set $\{(w_{1}, w_{2})\in W_{1}\cross W_{2}|w_{1}=w_{2}\}$ .Let $V=W\cross \mathbb{R}$ , $W_{1}\subset W\cross \mathbb{R}$, and the projection $W_{1}arrow W$ has non-zero degree. Here we assume $W$ and $W_{1}$ connected orientable manifolds ofthe same dimension. One can drop the orientability condition if works withcoefficient $\mathbb{Z}_{2}$ . Moreover let $W_{2}\subset W$ be acompact submanifold which lieson the left of $W_{1}$ , i.e., $W_{1}\cap\{(w_{2}, t_{2}+t) \in W\cross \mathbb{R}|(w_{2},t_{2})\in W_{2},t\leq 0\}=\emptyset$ .Theorem 3.2. If the projection of $\mathcal{L}_{2}(1)$ to $V$ lies on the right of the prO-jection $\mathcal{L}_{1}(1)$ , then we have$\#(\mathcal{L}_{1}(t)\mathrm{g}\mathcal{L}_{2}(t))reg\geq$ rank $H^{*}(W_{2})$ .The proofs of these theorems rely on the generating functions and thestable Morse theory132Postscript. In this lecture note we could note only apart of Eliashberg’stalk. He mentioned many other topics on symplectic field theory (SFT), sym-plectic cobordisms, compactness properties, generalized Viterbo’s theorem,Lagrangian skeletons, Lagrangian tori in $\mathrm{R}^{4}$ and so on.References[1] Y. Eliashberg, Topology of 2- note in $\mathbb{R}^{4}$ and symplectic geometry, TheFloer memorial volume, 335-353, 1995.[2] Y. Eliashberg and L. Polterovich, Local Lagrangian 2-knots are trivial,Ann. of math., 144 (1996), 61-76.[3] Y. Eliashberg and M. Gromov, Lagrangian intersection theory: Finite-dimensional approach, Amer. Math. Soc. Transl. (2) 186 (1998), 27-116133

Topology of Lagrangian Submanifolds (Proceedings of the Workshop "Algebraic Geometry and Integrable Systems related to String Theory")

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Topology of Lagrangian Submanifolds (Proceedings of the Workshop "Algebraic Geometry and Integrable Systems related to String Theory")

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