Asymptotic behavior of Vianna's exotic Lagrangian tori T a,b,c in CP 2 as a+b+c→∞
AASYMPTOTIC BEHAVIOR OF EXOTIC LAGRANGIAN TORI T a,b,c IN C P AS a + b + c → ∞ WEONMO LEE, YONG-GEUN OH, RENATO VIANNA
Abstract.
In this paper, we study various asymptotic behavior of the infinitefamily of monotone Lagrangian tori T a,b,c in C P associated to Markov triples( a, b, c ) described in [Via2]. We first prove that the Gromov capacity of thecomplement C P \ T a,b,c is greater than or equal to of the area of the complexline for all Markov triple ( a, b, c ). We then prove that there is a representativeof the family { T a,b,c } whose loci completely miss a metric ball of nonzero sizeand in particular the loci of the union of the family is not dense in C P . Contents
1. Introduction 22. Review of the exotic tori T a,b,c C P ( a , b , c ) as a symplectic reduction of C T -action on C P ( a , b , c ) and its moment polytope 72.4. Symplectic rational blow-down and almost toric fibration 112.5. Normalization of the polytope 143. Lower bound for the relative Gromov area 163.1. Almost toric blowup and symplectic balls 163.2. Method by Mandini and Pabiniak [MP] 184. Symplectic balls in the complement of T a,b,c in C P . 195. Geometry of the locus of the union of T a,b,c { T a,b,c } T a,b,c Mathematics Subject Classification.
Primary 53D05, 53D35.
Key words and phrases.
Vianna tori, Markov triple, orbifold projective plane, almost toric fibration,relative Gromov capacity, Lagrangian seeds.WL and YO were supported by the IBS project IBS-R003-D1. RV was supported by the Brazil’sNational Council of scientific and technological development CNPq, via the ‘bolsa de produtividade’fellowship, and by the Serrapilheira fellowship. a r X i v : . [ m a t h . S G ] J un WEONMO LEE, YONG-GEUN OH, RENATO VIANNA T a,b,c Introduction
In [Via1, Via2], the third named author constructed an interesting family ofinfinitely many monotone Lagrangian tori in C P by constructing a monotoneLagrangian torus associated to each of the Markov triples ( a, b, c ), i.e., positiveintegers satisfying the equation a + b + c = 3 abc. (1.1)For his construction, it was used constructions almost toric fibration, Symington’snodal surgery operations [Sym2] or the operation of rational blow down of theweighted projective planes C P ( a , b , c ) to C P . Denote by T a,b,c any realizationof the torus associated to the triple ( a, b, c ) in its Hamiltonian isotopy class in C P .To show that this family of T a,b,c are not pairwise Hamiltonian isotopic to oneanother, he used the disc-counting invariants which are known to be well-definedfor monotone Lagrangian tori [EP, Oh4].The starting point of our research in the present article lies in our attempt tounderstand the tori T a,b,c in terms of the geometry of Fubini-Study metric on C P .(We refer to Section 8 for more discussion on the related questions.) As a first steptowards this goal, we ask the following question Question 1.1.
Let { T a,b,c } be a fixed family of tori in C P . What is the geometricbehavior of these tori as a, b, c → ∞ ? For example, will the tori densely spread out C P as a + b + c → ∞ ?We denote by M the set of Markov triples. We remark that a specific constructionof the { T a,b,c } tori depends on various unspecified parameters. Because of the wayhow they are constructed, it is not easy to visualize the tori in the Fubini-Study metricof C P although they are well-defined up to Hamiltonian isotopy on ( C P , ω FS ).Fix any smooth metric on C P , e.g., take the Fubini-Study metric of C P . Question 1.2.
Let { T a,b,c } be any realization of the family of monotone Lagrangiantori in ( C P , ω FS ). Consider the following asymptotic quantity δ := inf ( a,b,c ) ∈ M sup x ∈ C P \ T a,b,c d ( x, T a,b,c ) (1.2)where d ( x, T a,b,c ) is the distance from x to T a,b,c . Is δ >
0? If so, estimate this δ .More intuitively and equivalently, the question asks if for any given point x ∈ C P and a positive constant δ >
0, there exists a Markov triple ( a, b, c ) such that B δ ( x ) ∩ T a,b,c (cid:54) = ∅ for the given family { T a,b,c } .This number δ is not a priori a symplectic invariant. More precisely if T (cid:48) a,b,c isanother realization of these tori, this quantity may vary. Because of this, we considerthe following quantity inf ( a,b,c ) ∈ M c G ( C P ; T a,b,c ) (1.3) XOTIC LAGRANGIAN TORI T a,b,c where c G ( C P ; T a,b,c ) is the relative Gromov area c G ( C P ; T a,b,c ) := sup e { πr | e : B ( r ) → C P \ T a,b,c is a symplectic embedding } (1.4)Relative Gromov area is a symplectic invariant and has been systematically studiedby Biran and Biran-Cornea [Bir1, Bir2, BC] for general pair of symplectic manifold( M, ω ) and its Lagrangian submanifold L . Remark 1.3.
We warn the readers that this definition of relative Gromov area isnot the one used by Biran and Cornea in [BC]. For our purpose in the present paper,we do not need their finer version and so we will just use the same term instead ofintroducing another different term for the Gromov area of the complement.The first theorem we prove in the present paper is the following rather optimallower bound. (See the construction given in Section 4.)
Theorem 1.4.
Let T a,b,c be any of monotone Lagrangian tori of [Via2] . Then inf ( a,b,c ) ∈ M c G ( C P ; T a,b,c ) ≥ π . Here we normalize the Fubini-Study form ω FS so that the area of the complex line is π . Motivated by the nature of our construction given in Section 4, we conjecturethat the equality holds in the above theorem.On the other hand, this lower bound is certainly not optimal for an individual torus.For example for the case of Clifford torus corresponding to ( a, b, c ) = (1 , , c G ( C P ; T , , ) ≥ π and Biran-Cornea [BC] proved c G ( C P ; T , , ) ≤ π , and hence c G ( C P ; T , , ) = 4 π . (1.5)This leads us to a very interesting open problem Problem 1.5.
Find the precise estimate of c G ( C P ; T a,b,c ) as done for the Cliffordtorus. The above theorem still does not prevent the loci of the union of the family { T a,b,c } being dense in C P , it may happen that there exists a family of symplecticballs of nonzero size associated to Markov triples ( a, b, c ) which are stretched thin andwildly spread around the ambient space C P without touching the corresponding T a,b,c torus respectively. Theorem 1.6.
There exists a family { T a,b,c } of tori that misses some closed metricball of non-zero size in C P . In fact, the supremum of the Gromov areas of suchmetric balls is π . In particular, the loci of the family is not dense in C P . The proof of this theorem will be given in Section 5 using the geometric mutationtheorem of the
Lagrangian seeds studied in in [STW] and [Ton, PT]. But one onlyneeds to look at Figure 9 to see how it goes.In the rest of the paper, we will provide various estimates relevant to the ballpacking problem in C P \ T a,b,c or ( C P \ E ) \ T a,b,c where E ⊂ C P is a smoothcubic curve, which corresponds to a Donaldson divisor of C P . One of the outcomesis the following result WEONMO LEE, YONG-GEUN OH, RENATO VIANNA
Theorem 1.7.
Any T a,b,c tori, in particular the Chekanov torus, can be embeddedinto the monotone C P k C P for k ≤ . This in particular affirmatively answers to a question posed by Chekanov andSchlenk [CS, Section 7] which asks whether Chekanov torus can be embedded into C P C P .In Section 8, we make further discussion and propose several open questionsrelated to the geometry of the tori T a,b,c .2. Review of the exotic tori T a,b,c The third named author [Via1, Via2] constructed a family of infinitely manynon-Hamiltonian isotopic monotone Lagrangian tori in C P as the transfers to C P of the fibers T ( a , b , c ) at the (labeled) barycenter of the moment polytope ofthe weighted projective plane C P ( a , b , c ) or its relevant almost toric fibration.He utilized Symington’s symplectic rational blow-down operations [Sym1] on eachneighborhood of orbifold points thereof and Moser’s deformation of the gluedsymplectic forms on the resulting blow-down to the Fubini-Study form on C P forhis construction. For the simplicity of notation, we denote by T a,b,c ⊂ C P any realization of the family of these tori in their Hamiltonian isotopy class in C P .We exclusively reserve T ( a , b , c ) for the fiber at the barycenter of the momentpolytope of C P ( a , b , c ). The torus T a,b,c can be also realized as the fiber of abase point of an almost toric fibration of C P [Via2]. An almost toric fibration is asingular Lagrangian fibration with nodal singular fibers. Here a nodal singular fibercarries an isolated singularity whose image under the almost toric projection lies onthe interior of the base diagram of the almost toric fibration. This image point iscalled a node .2.1. Almost toric fibration and nodal surgeries.
In this subsection, we recalldefinitions of almost toric fibration, nodal surgery operation and related resultsfrom [Via1, Section 2.3].
Definition 2.1 ([Zun1], [Via1]) . An almost toric fibration of a symplectic fourmanifold ( M, ω ) is a Lagrangian fibration π : ( M, ω ) → B such that any point of( M, ω ) has a Darboux neighborhood (with symplectic form dx ∧ dy + dx ∧ dy )in which the map π has one of the following forms: π ( x, y ) = ( x , x ) , regular point ,π ( x, y ) = ( x , x + y ) , elliptic, co-rank one ,π ( x, y ) = ( x + x , x + y ) , elliptic, co-rank two ,π ( x, y ) = ( x y + x y , x y − x y ) , nodal or focus-focus , with respect to some choice of coordinates near the image point in B . An almosttoric manifold is a symplectic manifold equipped with an almost toric fibration.A Lagrangian fibration induces an integral affine structure Λ on the base B withsingularity. Such pair ( B, Λ) is called an almost toric base [Sym2]. For an almosttoric manifold, there is a nontrivial monodromy around the nodal singular fiber.This prevents one from embedding the full almost toric base into ( R , Λ ) whereΛ is a standard integral affine structure. However removing an embedded curve R XOTIC LAGRANGIAN TORI T a,b,c joining a point of the boundary, in particular the vertex, of a moment polytope andthe node, called a branch curve , makes this embedding possible. Definition 2.2.
Suppose we have an integral affine embedding Φ : ( B − R, Λ) → ( R , Λ ), where ( B, Λ) is an almost toric base and R is a set of branch curves. A base diagram of ( B, Λ) with respect to R and Φ is the image of Φ decorated withthe following data: • an x marking the location of each node and • dashed lines indicating the portion of ∂ Φ( B − R ) that corresponds to R .If the direction of R is ( k, l ), then the monodromy around the node can berepresented by A ( k,l ) = (cid:18) − kl k − l kl (cid:19) with respect to some choice of basis. (See [Via1] for the details.)Consider a moment polytope in ( R ) ∗ . Nodal trade introduces a node and a cutinside the moment polytope in place of the vertex.(See the first two triangles inFigure 1) The corresponding fibration has a nodal singular fiber and becomes analmost toric fibration. Nodal slide literally “slides” a node along an eigenline of themonodromy map.(See the second and the third triangles in Figure 1) ++ Figure 1.
Almost toric surgeries on the moment polytope of C P Following Symington [Sym2], we consider the following operations on almosttoric fibrations which do not change the symplectic structure of the total space upto symplectomorphism.
Definition 2.3.
Let ( B, Λ i ) be two almost toric bases, i = 1 ,
2. We say that ( B, Λ )and ( B, Λ ) are related by a nodal slide if there is a curve γ in B such that • ( B − γ, Λ ) and ( B − γ, Λ ) are isomorphic, • γ contains one node of ( B, Λ i ) for each i and • γ is contained in the eigenline (line preserved by the monodromy) throughthat node. Definition 2.4.
Let ( B i , Λ i ) be two almost toric bases, i = 1 ,
2. We say that( B , Λ ) and ( B , Λ ) differ by a nodal trade if each contains a curve γ i starting at ∂B i such that ( B − γ , Λ ) and ( B − γ , Λ ) are isomorphic, and ( B , Λ ) has oneless vertex than ( B , Λ ).It is shown by Symington [Sym2] that these almost toric operations do not changethe diffeomorphism type they represent and keep the symplectic structure up toisotopy and so Moser’s argument shows that the two almost toric manifolds beforeand after the operations are symplectomorphic. WEONMO LEE, YONG-GEUN OH, RENATO VIANNA C P ( a , b , c ) as a symplectic reduction of C . The base diagram foran almost toric fibration of C P having a monotone Lagrangian torus T a,b,c atits barycenter looks like the moment polytope of a weighted projective plane C P ( a , b , c ), except that the diagram is equipped with nodes and cuts.In this subsection we describe various aspects of geometry of C P ( a , b , c ) as atoric orbifold. We consider the following C ∗ action of S on C \ { } defined by ζ · ( x, y, z ) := ( ζ a x, ζ b y, ζ c z ) (2.1)for ζ ∈ C \
0. The weighted projective plane C P ( a , b , c ) with weights ( a , b , c )as a complex orbifold is nothing but the quotient of ( C ∗ ) by this action. We denoteby [ x : y : z ] its element represented by ( x, y, z ) ∈ C \ { } . C P ( a , b , c ) is anorbifold with three orbifold points [1 : 0 : 0] , [0 : 1 : 0] , [0 : 0 : 1] . Their correspondingorbifold structure groups are Z /a Z , Z /b Z , Z /c Z , respectively.For our purpose of proving Main Theorem, we need to explicitly express therelevant symplectic structure and T -action on C P ( a , b , c ) starting from the linearsigma model construction as in [Wit, Aud], which was exploited in the LagrangianFloer theory of toric manifolds in [CO, Section 3]. We regard the toric orbifold C P ( a , b , c ) as the symplectic reduction φ − K ( r ) /K of C under the action of thecircle subgroup K ⊂ T for a suitable choice of r ∈ R where φ K : C → R is theassociated moment map. Then it carries the canonical action of the residual torus T /K ∼ = T thereon. We will call this particular T -action the residual T -action on φ − K ( r ) /K ∼ = C P ( a , b , c ).Now we need to describe this torus action on C P ( a , b , c ) and its momentpolytope explicitly, employing notations from [Abr, Section 2.2]. For this purpose,we start with the standard torus action of T on (cid:16) C , (cid:80) d =3 i =1 du i ∧ dv i (cid:17) defined by θ · ( z , z , z ) = ( e iθ z , e iθ z , e iθ z ) . This T action has its moment map φ T : C → R given by φ T ( z , z , z ) = (cid:88) i =1 | z i | e ∗ i + λ for an arbitrary choice of constant vector λ ∈ ( R ) ∗ in general where { e ∗ , e ∗ , e ∗ } is thebasis of ( R ) ∗ dual to the standard basis { e , e , e } of R . Setting λ = (cid:80) i =1 λ i e ∗ i with λ = − b c , λ = λ = 0we have φ T ( z , z , z ) = (cid:18) | z | − b c (cid:19) e ∗ + | z | e ∗ + | z | e ∗ . Being a subgroup, any circle subgroup K ⊂ T naturally acts on C with momentmap φ K = ι ∗ ◦ φ T = (cid:88) i =1 (cid:18) | z i | λ i (cid:19) ι ∗ ( e ∗ i ) ∈ k ∗ . Then the symplectic quotient M P := φ − K (0) /K of C carries the canonical reducedsymplectic form and the residual torus action by T /K ∼ = T whose moment map XOTIC LAGRANGIAN TORI T a,b,c image is the labeled polytope P described in subsection 2.3. Furthermore thisreduced space is precisely the symplectic orbifold C P ( a , b , c ) equipped with the2-torus action by the torus T ∼ = T /K .We now identify what this circle subgroup K ⊂ T associated to C P ( a , b , c )is. Define a linear map β : R → R by β ( e i ) = m i µ i for i = 1 , , k the kernel of β . We have the short exact sequences0 → k ι → R β → R → → ( R ) ∗ β ∗ → ( R ) ∗ ι ∗ → k ∗ → . Denote by K ⊂ T the subgroup generated by k . β also induces the exact sequenceof abelian groups 0 → K → R / (2 π Z ) → R / (2 π Z ) → . For any element θ = ( θ , θ , θ ) in k , since m i = 1 for all i = 1 , , m i θ i µ i = Σ θ i µ i ∈ (2 π Z ) . A simple computation shows
Lemma 2.5.
The one-dimensional integral sub-lattice k is generated by ( a , b , c ) .Proof. Any element θ = ( θ , θ , θ ) in k satisfiesΣ θ i µ i = ( − ( bl − θ − ( al − θ + θ , − b θ + a θ ) ≡ (0 , . From the second slot, θ = b a θ . This implies θ = ( bl − θ + ( al − b a θ = θ a ( a bl − a + ab l − b )= c a θ . Here the last equality comes from equalities a bl − a + ab l − b = ab ( al + bl ) − a − b and (2.5) and the fact that ( a, b, c ) is a Markov triple. Therefore k is generated by( a , b , c ). (cid:3) Therefore this lemma is consistent with the C ∗ action (2.1).2.3. Residual T -action on C P ( a , b , c ) and its moment polytope. Usingthe generalization of Delzant’s argument applied to toric orbifold, Lerman andTolman [LT] describe its associated orbifold moment map φ P and the associatedsymplectic form ω P on φ − P (Int P ) explicitly, which we now recall. In order to applytheir argument, we borrow relevant definitions and arguments from the expositionof Abreu [Abr, Section 2.2] now. Definition 2.6.
A convex polytope P in ( R n ) ∗ is said to be simple and rational if • n facets meet at each vertex p , WEONMO LEE, YONG-GEUN OH, RENATO VIANNA • those edges meeting at the vertex p are all rational, i.e., each edge has theform p + tv i where 0 ≤ t ≤ ∞ , v i ∈ ( Z n ) ∗ and • the v , · · · , v n can be chosen to be a Q − basis of the lattice ( Z n ) ∗ . We call the polytope P a labeled polytope if it is a rational simple convex polytopeand there is a positive integer label on the interior of each facet. Theorem 2.7. [LT, Abr]
Let ( M, ω ) be a compact symplectic toric orbifold, withmoment map φ : M → ( R n ) ∗ . Then P = φ ( M ) is a labeled polytope. For each facet F of P , there exists a positive integer m F , the label of F , such that the orbifoldstructure group of every point in φ − (Int F ) is Z /m F Z . Two compact symplectic toric orbifolds are equivariantly symplectomorphic if andonly if their associated labeled polytopes are isomorphic. Moreover, every labeledpolytope arises from some compact symplectic toric orbifold.
The following lemma computes the labels of the facets of the moment polytopeof C P ( a , b , c ), which plays an important role in our proof. Lemma 2.8.
Denote by P the moment polytope of C P ( a , b , c ) of the associatedresidual T -action. Then the label m F on every facet F of P is 1.Proof. Recall that the orbifold structure group of a point [ x : y : z ] in the weightedprojective plane C P ( w , w , w ) is Z /g Z where g is the greatest common divisorof those weights whose component is non-zero. For a point [ x : y : z ] with x (cid:54) = 0and y (cid:54) = 0, gcd of a and b is 1. Recall that a, b, c are mutually coprime. (See[Via2, Proposition 2.2].) Therefore at those points the orbifold structure groupsare trivial. Similarly every point [ x : y : z ] but three orbifold points has a trivialorbifold structure group. Therefore each point fibering over the interior of eachfacets has a trivial orbifold structure group, which means the point is smooth. Inother words, the labels of the other facets are 1. This finishes the proof. (cid:3) Finally we find an explicit coordinate formula of the moment map φ P : M P → ( R ) ∗ . Let π : φ − K (0) → M P be the quotient map and i : φ − K (0) → C the inclusionmap. Then by definition of the moment map [MW], the associated moment map φ P : M P → ( R ) ∗ satisfies the following equation β ∗ ◦ φ P ◦ π = φ T ◦ i. (2.2)Let ( x, y ) (row vector) be the coordinate of ( R ) ∗ and ( z , z , z ) (column vector)be the complex coordinate of C . With respect to the standard basis, the map β : R → R can be written as β = (cid:18) − ( bl − − ( al −
1) 1 − b a (cid:19) . Substituting ( z , z , z ) ∈ φ − T (0) into the left hand side of (2.2) and setting( x, y ) = φ P ( π ( z , z , z )) , we get | z | − b c | z | | z | = β ∗ (cid:18) xy (cid:19) = − ( bl − x − b y − ( al − x + a yx XOTIC LAGRANGIAN TORI T a,b,c By equating the first and the last terms of the equation and solving it for ( x, y ),we obtain the coordinate formula of the associated moment map φ P whose value at[ z : z : z ] ∈ C P ( a , b , c ) is given by( x, y ) = (cid:32) | z | ( abc ) a | z | + b | z | + c | z | , (cid:0) | z | + ( al − | z | (cid:1) ( bc ) a | z | + b | z | + c | z | (cid:33) (2.3)for all ( z , z , z ) ∈ φ − T (0) ⊂ C . This ends our symplectic description of sym-plectic orbifold C P ( a , b , c ), the T -action and its associated moment map φ P : C P ( a , b , c ) → ( R ) ∗ . Remark 2.9.
Note that C P ( a , b , c ) is covered by three orbifold charts. Forexample, { [ x : y : z ] | x (cid:54) = 0 } is one of them. This is homeomorphic to C /µ a where the group action is given by ζ · ( y, z ) = ( ζ b y, ζ c z ) i.e., its associatedweights are given by a ( b , c ). Denote this orbifold chart as U a . Similarly, denote U b = { [ x : y : z ] | y (cid:54) = 0 } and U c = { [ x : y : z ] | z (cid:54) = 0 } . It is easy to check that oneach orbifold chart, the map φ P in (2.3) is invariant under the corresponding groupaction. Thus φ P is well-defined.Three orbifold points [1 : 0 : 0] , [0 : 1 : 0] , [0 : 0 : 1] are mapped to the verticesopposite to the edges associated to a u , b u , c u , respectively. One can checkthat points of the form [0 : y : z ] ∈ C P ( a , b , c ) fibers over the points contained inthe edge corresponding to a u . Similarly points of the form [ x : 0 : z ] , [ x : y : 0]fibers over points in the edge b u , c u , respectively.We now visualize the image in ( R ) ∗ of the moment map φ P . First we recall thatevery convex polytope P can be written as the intersection of a finite number oforiented half-spaces. To define a labeled polytope, we attach a label m i on the i -thfacet and consider the intersection P = (cid:92) i =1 { ( x, y ) ∈ ( R ) ∗ : L i ( x, y ) = (cid:104) ( x, y ) , m i µ i (cid:105) − λ i ≥ } where µ i is the i -th inward primitive integral vector normal to the i -th facet and λ i is areal number.First we recall the polytope considered in [Via2] for his construction of T a,b,c .In the above expression of general P , we consider the edge of affine length a , b ,respectively, as the first, second facet. Then we get µ = ( − ( bl − , − b ) , µ = ( − ( al − , a ) , µ = (1 , L ( x, y ) := − ( bl − x − b y + b c ≥ L ( x, y ) := − ( al − x + a y ≥ L ( x, y ) := x ≥ λ = − b c , λ = λ = 0. We denote the polytope given by this equation by P = P ( a , b , c ) . (2.4)This is precisely the one used in [Via2]. See Figure 2.The following proves that the labeled polytope P of the toric orbifold C P ( a , b , c )is exactly the moment polytope described above equipped with label 1 on the interiorof each facet. a u b u c u Figure 2.
Moment polytope of C P ( a , b , c ) Proposition 2.10.
The moment image of φ P : C P ( a , b , c ) → ( R ) ∗ given aboveis (2.4) .Proof. It is straightforward to check from the explicit formula (2.3). (cid:3)
As described in [Via2], its moment polytope under the residual torus action isthe triangle with edges parallel to the vectors a u , b u , c u where they satisfythe balancing condition a u + b u + c u = (0 , u = ( b , − ( bl − , u = − ( a , al − , u = (0 , a u + b u + c u = (0 , , we obtain a relation al + bl = 3 c. (2.5)Here each of l , l , l is a positive integer coprime to a, b, c respectively. Each ofthese integers can be realized as a winding number of some section of a trivializationover the boundary of unit disk appearing in the definition of Lagrangian pinwheel ,embedded in C P . (See [Kho, ES].) Furthermore, Evans and Smith [ES] provedthat the integers satisfy the following congruence relations; l ≡ − a ) l ≡ − b ) bl ≡ ± c (mod a ) cl ≡ ± b (mod a ) al ≡ ± c (mod b ) cl ≡ ± a (mod b )Then using the coordinates of ( R ) ∗ , l i ’s could be computed explicitly from Figure4 below, and following a series of nodal surgery operations followed by transferring XOTIC LAGRANGIAN TORI T a,b,c the cut operation, starting from the moment polytope of C P which corresponds toMarkov triple (1 , , Remark 2.11.
When deriving (2.3), using − ( bl − x − b y = | z | − b c , we getanother expression for y = c − ( | z | + ( bl − | z | )( ac ) a | z | + b | z | + c | z | . Then one can easily check that using coordinates on orbifold charts U a , U b , U c the image of φ P is exactly P . In other words, on U a , its image point ( x, y ) of (2.3)satisfies L ( x, y ) > , L ( x, y ) ≥ , L ( x, y ) ≥ . On U b , the image point ( x, y ) satisfies L ( x, y ) ≥ , L ( x, y ) > , L ( x, y ) ≥ . On U c , the image point satisfies L ( x, y ) ≥ , L ( x, y ) ≥ , L ( x, y ) > . The associated symplectic form ω P can be written explicitly in terms of labelledpolytope data using the analogue of Guillemin’s formula [Gui]. Theorem 2.12. [CDG]
Let P be a labelled polytope as above and d be the numberof facets of P . Define a function L ∞ ( u ) = d (cid:88) i =1 (cid:104) u, m i µ i (cid:105) (2.6) on ( R n ) ∗ . We have ω P = √− ∂∂φ ∗ P (cid:32) d (cid:88) i =1 λ i log L i + L ∞ (cid:33) (2.7) on φ − P (Int P ) . In our case, d = 3 and n = 2. For ( X, Y ) in ( R ) ∗ , L ∞ ( X, Y ) = (cid:104) ( X, Y ) , (cid:88) i =1 µ i (cid:105) = [3 − ( al + bl )] X + ( a − b ) Y. Symplectic rational blow-down and almost toric fibration.
One real-ization of the T a,b,c tori is given as the barycentric fibers of various almost toricfibrations on C P depending on ( a, b, c ). As mentioned before, its base diagram( B, Λ) looks like the moment polytope P = P ( a , b , c ) of C P ( a , b , c ) exceptthat there is a node at each vertex with a cut in the direction of the correspondingnode.Each small neighborhood of a vertex can be realized as an almost toric base withone node whose boundary is a lens space of the form L ( k , kl − . Let us describea smooth 4 dimensional manifold fibering on this neighborhood, the correspondingbase in P and its symplectic rational blow-down surgery. Let k, l be a pair of coprime positive integers. Consider the base diagram givenby the open subset U k,l ⊂ ( R ) ∗ consisting of the points ( x, y ) contained in theintersection of the half-spaces y ≥ kl − k x, x ≥ , y > , which is bounded by an arbitrary embedded arc γ joining two points on the edgesrespectively contained in the two lines y = kl − k x , x = 0. It also carries a node on acut in the direction of the vector ( k, l ). + k k l kl − Figure 3.
Almost toric base U k,l A smooth 4 dimensional manifold B k,l that fibers over U k,l is a rational homologyball whose boundary is a lens space L ( k , kl − . This lens space fibers over theembedded arc γ [Sym2, Section 9.3] contained in U k,(cid:96) .On the other hand, it is proved in [Via2, Proposition 2.2] that the boundary of aneighborhood of the orbifold point projected to the vertex opposite to a u , b u , c u is the lens space of the form L ( a , al − , L ( b , bl − , L ( c , cl − , respectively. Here each l i for i = 1 , , a, b, c ,respectively. Now we consider a small neighborhood N a of an orbifold point whichfibers over the vertex opposite to the edge a u . As two collar neighborhoods of aboundary of N a and of B a,l fiber over the same simply connected base, there existsa symplectomorphism ψ a from a collar neighborhood of the boundary of N a to acollar neighborhood of the boundary of B a,l . Similar results hold for the remainingneighborhoods N b and N c .(See [Sym1])Then the rational blow-down surgery replaces a neighborhood N a , N b , N c of an orbifold point by a rational homology ball B a,l , B b,l , B c,l , respectively.Furthermore, the symplectic structures ω P and ω a,l , ω b,l , ω c,l induce a symplecticstructure as follows.As performed in [Via2, Corollary 2.5], applying rational blow-down on eachneighborhood of the three orbifold points of C P ( a , b , c ) yields an almost toricfibration of C P . We denote by C P a,b,c the total space of this almost toric fibrationfor some base ( B, Λ). More precisely, applying the rational blow-down surgery oneach neighborhood of three orbifold points in C P ( a , b , c ), we obtain an almosttoric manifold C P a,b,c symplectomorphic to C P C P a,b,c := (cid:0) C P ( a , b , c ) \ ( N a ∪ N b ∪ N c ) (cid:1) (cid:91) ψ ( B a,l ∪ B b,l ∪ B c,l ) XOTIC LAGRANGIAN TORI T a,b,c equipped with an interpolated symplectic form (cid:101) ω = (1 − χ a − χ b − χ c ) ω P + χ a λ a ω a,l + χ b λ b ω b,l + χ c λ c ω c,l . (2.8)Here each χ a is a cut-off function which is 1 on the corresponding rational homologyball and 0 otherwise. And λ a , λ b , λ c are suitably chosen positive real-valued functionsso that (cid:101) ω becomes nondegenerate and closed.Corresponding to base diagram ( B, Λ), P has three nodes with eigenrays [Via2])pointing towards ( a, l ) , ( b, − l ) , ( − ( a + b ) , bl − al )issued at the vertices opposite to a u , b u , c u , respectively. The three eigenraysmeet at one point, the labeled barycenter , in the interior of P . (See Figure 4.Red dashed lines represent respective cuts and the blue dot represents the labeledbarycenter.)Following [Via2, Remark 2.3], we set w = − ( a, l ) , w = ( − b, l ) and w as thevectors representing cuts. They satisfy relations acw − bcw c u (2.9) bcw − abw b u (2.10) abw − acw a u (2.11)and abc aw + bw + cw ) = 0 . (2.12)From (2.10), we have aw = cw − bu = ( a (3 ab − c ) , b ( al − − cl ). The slopeof w is then l (3 ab − c ) − ba (3 ab − c ) = l a − ba a + b c = l a − bca ( a + b ) . By (2.11), the slope of w is ( c − ab ) l + 3 ab (3 ab − c ) . (2.13)Also from (2.12), the slope of w can be written as al − bl a + b . (2.14)These three expressions of the slope of w are indeed the same.By translation, we may identify the lower left vertex of P with the origin in( R ) ∗ . Consequently vertices opposite to edges b u , c u respectively are located atthe points (0 , c ) , ( a b , c − a ( bl − . Then we derive ( a b , c − a ( bl − a b , b ( al − ( abc , bcl ) ( a b , b ( al − , , c )++ + Figure 4.
The polytope P Lemma 2.13.
Let P be the polytope associated to the above base diagram ( B, Λ) .Then the torus T ( a , b , c ) is located at the point (cid:18) abc , bcl (cid:19) ∈ Int P ( a , b , c ) . (2.15) Proof.
Using the slope formula we derived above, we easily check that the threeeigenrays are given by y = l a x,y = c − l b x,y = b ( al −
1) + (cid:20) l a − bca ( a + b ) (cid:21) ( x − a b )and that they meet at one point (cid:18) abc , bcl (cid:19) . (cid:3) Normalization of the polytope.
When we choose χ i ’s and λ j ’s in (2.8), werequire the (volume) normalization condition in addition so that (cid:90) C P a,b,c (cid:101) ω = (cid:90) C P ω . Such a choice can be always made by suitably choosing the functions λ i , and thenMoser’s deformation trick produces a diffeomorphism between the two symplecticforms (cid:101) ω and ω FS α : ( C P , ω FS ) → ( C P a,b,c , (cid:101) ω )such that α ∗ (cid:101) ω = ω FS . We fix such a symplectic (actually K¨ahler) form (cid:101) ω . (See[Via1] for more detailed explanation.)To apply the above mentioned Moser’s deformation to the pair of forms α ∗ (cid:101) ω and ω FS , we need to suitably normalize the size of the polytope P ( a , b , c ) so XOTIC LAGRANGIAN TORI T a,b,c that the cohomology classes [ α ∗ ( (cid:101) ω )] , [ ω FS ] ∈ H ( C P , R ) should be the same. Sincedim R H ( C P , R ) = 1, we know [ α ∗ ( (cid:101) ω )] = C [ ω FS ] for some positive constant C > C is. We first recall that monotonicity constants ofall monotone Lagrangian tori in C P are the same. For example, the Maslov index2 discs of T a,b,c in C P have the same symplectic areas independent of a, b, c .On the other hand, by the classification theorem from [CO], there exist threeobviously seen Maslov index 2 holomorphic discs attached to T ( a , b , c ) which areassociated to the facets of the polytope P . Denote these holomorphic discs by D ( µ i )for i = 1 , , i -th facet of P. The following fact is stated in [Via2, Paragraph after Prop. 2.4] without proof.Because this is an important element in our study of the present paper, we give itsproof for readers’ convenience.
Proposition 2.14.
Consider the polytope P = P ( a , b , c ) described around (2.4) .If we scale P by dividing by abc , then the α ∗ ( (cid:101) ω ) -symplectic areas of Maslov index2 holomorphic discs attached to T a,b,c are the same as that of ω FS . In particular, [ α ∗ ( (cid:101) ω )] = [ ω FS ] for all a, b, c .Proof. We start with the following area formula for holomorphic disc of Maslovindex 2 from [CO] for general toric manifolds.
Lemma 2.15 (Theorem 8.1 [CO]) . Let ω X be the toric symplectic form, which isthe reduced form of the standard symplectic form ω on C N under the linear sigmamodel construction. Consider the residual T n action on X and its moment map π : X → t ∗ ∼ = R n . Let L = π − ( A ) be the fiber torus based at A ∈ Int P . Then thesymplectic area of the holomorphic disc corresponding to the i-th facet is given by π ( (cid:104) A, µ i (cid:105) − λ i ) . Using this area formula, we computearea D ( µ ) = 2 π ( (cid:104) A, µ (cid:105) + b c )= 2 π (cid:18) − abc bl − − bcl b + b c (cid:19) = 2 π bc (cid:0) bc − a ( bl − − b l (cid:1) = 2 π bc (3 bc + a − b ( bl + al ))By (2.5), we compute area D ( µ ) = 2 π abc. Similarly, for the second facet,area D ( µ ) = 2 π ( (cid:104) A, µ (cid:105) )= 2 π (cid:18) − abc al −
1) + bcl a (cid:19) = 2 π abc. For the third facet, area D ( µ ) = 2 π ( (cid:104) A, µ (cid:105) ) = 2 π abc. (This explanation shows that every holomorphic disc has the same area as it shouldbe because because Lagrangian tori T ( a , b , c ) are monotone.) Therefore after ifwe scale the symplectic form (cid:101) ω by abc , all holomorphic discs of Maslov index 2 hasarea π which is the same as that of Clifford torus of in C P with respect to ω FS .Obviously this area is independent of the choice of Markov triple ( a, b, c ). (cid:3) Lower bound for the relative Gromov area
Let ( M n , ω ) be a symplectic manifold. We recall the definition of Gromov area .Denote B n ( d ) = { z ∈ C n : | z | ≤ d } for d > Definition 3.1 (Relative Gromov area) . Let L ⊂ M be a compact subset. Considera symplectic embedding e : B n ( d ) → M \ L . The relative Gromov area is definedby c G ( M ; L ) := sup e { πd | e : B n ( d ) → M \ L is a symplectic embedding } We are interested in studying the behavior c G ( C P ; T a,b,c ) as max { a, b, c } → ∞ .For this purpose, we first recall two methods of finding a symplectic balls in thecontext of toric manifolds.The first one is given by Karshon [Kar] who uses the shape of triangle and theother is given by Mandini and Pabiniak [MP] who uses the shape of diamond in themoment polytope.3.1. Almost toric blowup and symplectic balls.
In this subsection, we followthe exposition given in [Via3, Section 2.4] on the almost toric blowup to which thereaders are referred for details. See also [Zun2, Sym2, LS].In short, the picture below describes an almost toric blowup. The exceptionalcurve lies over the dashed line, consisting of one circle on each fibre that collapsesas it approaches the edge and as it approaches the node.
Figure 5.
Almost toric blowup.In particular, one must be able to find a symplectic ball of Gromov area π(cid:15) , in aneighbourhood of an affine triangle in an ATF (almost toric fibration), correspondingto the missing triangle after the almost toric blowup.More precisely, suppose we see a triangle inside a toric region of an ATF, asillustrated in Figure 6. Viewing the preimage of any small neighborhood of this XOTIC LAGRANGIAN TORI T a,b,c triangle as a blowdown of the corresponding neighborhood of the exceptional curveas in Figure 5, we can then infer that there is a symplectic ball of capacity π(cid:15) projecting into the preimage of this neighborhood of the triangle. Figure 6.
A symplectic ball B of capacity smaller than π(cid:15) projects into the shaded triangle.So for our purpose, every time we encounter a triangle as in Figure 6, we knowthere is a symplectic ball B of any given capacity smaller than π(cid:15) that projectsinto the shaded triangle.Just for visualization purpose, we describe how a symplectic ball B centred at apoint lying over an edge of an ATF, and projecting into the triangle of Figure 6,should look like.We first describe how its boundary ∂B ∼ = S intersect each fibre of the triangle.To ensure that the topological sphere described below is indeed the boundary of asymplectic four ball, one needs to make more specific choices for these intersections.We omit these details, since by the previous discussion, we do not really need them.The boundary ∂B ∼ = S intersects the corresponding triangle in the toric sectorof the ATF as follows:a) Consider the edge of the triangle corresponding to the intersection withedge of the ATF. The circle fibres, corresponding to the relative interior ofthe edge, intersect ∂B in two points. The circles fibering over the endpointsof the edges, intersect ∂B in one point each. Hence, ∂B intersects the fibreover the edge in a circle.b) The tori living over the remaining edges of the triangle intersect ∂B inone circle. The class of this circle is the collapsing class associated to theedge. So the circles collapse to a point as we approach the edge, which isconsistent with the previous item a).c) The tori living over the interior edge, intersect ∂B in two circles, also in thecollapsing class associated to the edge. Hence, they collapse to two points aswe approach the edge of the item a). If we approach the fibres of describedin item b), these two circles collapse to the corresponding single circle.In particular, consider a segment in the triangle, parallel to the edge of the ATF,connecting points of the edge of the triangle. It divides the triangle in two parts,the top part being a similar triangle. The intersection of ∂B with the fibres overthis segment form a torus. The top part then intersects ∂B in a solid torus, wherethe family of tori living over the corresponding parallel edges collapse to the circleliving over the vertex. The bottom part also intersects ∂B in a solid torus, now thecorresponding parallel segments converge to the edge of the intersecting the edge ofthe ATF, whose fibres intersect ∂B in a circle, as in item a) above. It is easy to see that this circles correspond to generators of H of the torus living over our initialsegment. Hence, we have indeed ∂B ∼ = S .To get B , we consider S ’s as above, projecting to smaller triangles similar toeach other, eventually collapsing to a point in the middle of the edge of the startingtriangle.One is able to see that we can get balls of any capacity smaller than π(cid:15) , projectinginside our given triangle. In particular, we can get a symplectic ball of capacity π(cid:15) ,if we get this triangle inside a slightly bigger one in our ATF.3.2. Method by Mandini and Pabiniak [MP] . We first recall a result by Man-dini and Pabiniak [MP]. Following [MP], we consider the subset ♦ ( d ) = { ( x, y ) ∈ R : | x | + | y | < d } . Proposition 3.2. [MP]
For each ε > the 4-ball B ( (cid:112) d − ε )) of capacity π ( d − ε ) symplectically embeds into ♦ ( d ) × (0 , π ) ⊂ R × R . Therefore, if for atoric manifold ( M , ω ) with moment map φ , Ψ( ♦ ( d )) + x ⊂ Int φ ( M ) for some Ψ ∈ GL (2 , Z ) and x ∈ R , then the Gromov area of ( M , ω ) is at least πd . Construction of such a symplectic embedding of a 4-ball B ( (cid:112) d − ε )) into ♦ ( d ) × (0 , π ) ⊂ R × R is given in the proof of [LMS, Lemma 4.1] using [Sch,Lemma 3.1.8] which is irrelevant to the toric structure of a symplectic manifold. They use only an area-preserving map from a 2-ball of capacity d to rectangle R ( d ) = (0 , d ) × (0 , R ( d )is equal to the area enclosed by a concentric circle in the 2-ball, in R ( d ).Let ( M , ω ) be a toric symplectic manifold. If an affine transformation maps adiamond ♦ ( d ) into the interior of φ ( M ) , then some subset of φ − ( ♦ ( d )) is symplec-tomorphic to ♦ ( d ) × (0 , π ) . Here we use the identification S = R / π Z . Then theabove symplectic embedding of the diamond induces a symplectic embedding of B ( (cid:112) d − ε )) into ( M , ω ) . Adopting the same idea in the almost toric case, not only for toric manifoldsbut also for almost toric manifolds, Proposition 3.2 will hold the case with suitablemodifications.
Proposition 3.3.
Let ( M, ω ) be an almost toric manifold with almost toric fibration π : ( M, ω ) → B. If Ψ( ♦ ( d )) + x ⊂ Int π ( M ) \ { nodes } for some Ψ ∈ GL (2 , Z ) and x ∈ R , then the Gromov area of ( M , ω ) is at least πd. Proof.
A crucial ingredient in the toric case exploits the fact that each fiber over aninterior point of its moment polytope is a 2-torus and we have φ − ( ♦ ( d )) ∼ = ♦ ( d ) × T symplectically via the action-angle variables. Let π : ( M, ω ) → B be an almost toricfibration. In the almost toric base a smooth fiber over its interior point is a 2-torusaway from the singular values of π .Assume that there is an affine transformation Φ of ♦ ( d ) mapping into Int π ( M ) \{ nodes } . Then some subset of π − (Φ( ♦ ( d ))) ⊂ M \ { nodal fibers } is symplecto-morphic to ♦ ( d ) × (0 , π ) . Symplectic embedding of the 4-ball B ( (cid:112) d − ε )) ofcapacity 2 π ( d − ε ) into ♦ ( d ) × (0 , π ) ⊂ R × R in [LMS] completes the proof. (cid:3) XOTIC LAGRANGIAN TORI T a,b,c Remark 3.4.
Recall that both nodal trade and nodal slide operations induce twodiffeomorphic smooth 4-manifolds with isotopic symplectic forms. Performing nodalslide of a node towards either the vertex or the barycenter of a base diagram alongeach eigenray allow us to find the size “ d ” of a diamond ♦ ( d ) while avoiding all nodesinside the base diagram. Since Gromov area is invariant under a symplectomorphism,combining these, we can find a maximal lower bound for the Gromov area of analmost toric manifold.4. Symplectic balls in the complement of T a,b,c in C P . By the discussion given in Subsection 3.1, in order to see a symplectic ball inthe complement of T a,b,c , it is enough to identify a corresponding triangle with oneside in the edge of the ATF base diagram, or a diamond in the interior of the baseavoiding T a,b,c .For the simplicity of the constants appearing in this section, we scale the sym-plectic form so that the area of the Maslov index 2 disk is 1 or the area of the lineis 3. Therefore, each side of the toric diagram has length 3, corresponding to thearea of the line and the area of the anti-canonical divisor is 9, since it has degree 3.Therefore, the total area of the boundary of any ATF base diagram is 9. Moreoverlooking at the orbifold limit the largest edge has length ≥ monotone triangle , a triangle with one edge at the boundary of thebase diagram of the ATF, with all the affine lengths of the edges being 1, andthe associated symplectic ball a monotone ball . Inside a neighbourhood of thistriangle projects a ball of capacity one that endows the monotone symplectic formin C P C P after blown down. Figure 7.
A symplectic ball in the complement of the monotonefibre in an ATF of C P projects into a arbitrarily small neighbour-hood of the monotone triangle (shaded). Theorem 4.1.
Rescale the standard Fubini-Study form so that the area of the lineis π so that the Maslov index 2 disk has area π/ . Then inf ( a,b,c ) ∈ M c G ( C P ; T a,b,c ) ≥ π Proof.
Up to SL (2 , Z ), we can always take the largest edge of ATF base diagramassociated to each Markov triple ( a, b, c ) to be horizontal, with one of the cutsintersecting that edge vertical, as illustrated in Figure 7. Hence the monotonefibre T a,b,c is at height 1. Therefore we can find a equilateral right triangle so that one of the vertices thereof is put right at the point. Because the affine lengthof the horizontal edge is ≥ T a,b,c . This finishes the proof. (cid:3) Even though our construction indicates the lower bound given in the abovetheorem may be optimal, it is not clear whether it is indeed the case. In fact,as already mentioned in the introduction, the lower bound will be bigger for anindividual torus, since we can get a neighbourhood of the monotone triangle – seeFigure 7 again. In other words, c G ( C P ; T a,b,c ) ≥ π + ε a,b,c , for some ε a,b,c > c G ( C P ; T text Cl ) = π for the Clifford torus T Cl ∼ = T , , in [BC].However we conjecture the above lower bound is indeed optimal. Conjecture 4.2.
There is no monotone ball in the complement of T a,b,c , and inf ( a,b,c ) ∈ M c G ( C P ; T a,b,c ) = 2 π with the convention of π being the capacity of the monotone ball. We note that the ball presented in this theorem intersects the ‘boundary divisor’by construction. Denote by E the preimage of the boundary of the base diagram.So we consider C P \ E , the complement of E (still endowed with the finite volumeform coming from C P , i. e., without completing it). Figure 8.
A ball of capacity π in the complement of the mono-tone fibre in an ATF of C P \ E projects into a arbitrarily smallneighbourhood of the diamond ♦ ( / ) (shaded).Looking again back at Figure 7, we can indeed see a diamond ♦ ( / ) in thecomplement of E . A small neighbourhood thereof contains a ball of capacity π .(See Figure 8.) An application of Proposition 3.3 gives rise to the following initialestimate inf ( a,b,c ) ∈ M c G ( C P \ E ; T a,b,c ) ≥ π . (4.1)We will give a better improved estimate later in Section after a finer study of thebase diagram associated to ( a, b, c ).5. Geometry of the locus of the union of T a,b,c In this section, we construct a representative of the family { T a,b,c } of monotoneLagrangian tori such that the loci of the tori T a,b,c is not dense in C P . XOTIC LAGRANGIAN TORI T a,b,c Construction of a non-spreading family { T a,b,c } . We consider the config-uration formed by the union of the Clifford torus and three Lagrangian disks, as firstexposed in [STW], see also [Ton, PT]. We will construct a family such that all tori T a,b,c reside in an arbitrarily small neighbourhood of the locus of this configuration.We denote L Sk for this configuration, which can also be thought as a Lagrangianskeleton of the Liouville domain M = C P \ E . This Lagrangian skeleton consistsof the monotone Clifford torus T Cl together with three Lagrangian disks, withboundaries on T Cl .To make our construction in perspective, we recall the notion of Lagrangian seeds from [PT].
Definition 5.1 (Definition 4.7 [PT]) . A Lagrangian seed ( L, { D i } ) in a symplectic4-manifold X consists of a monotone torus L ⊂ X , and a collection of embedded La-grangian disks D i ⊂ X with boundary on L , which satisfies the following conditions.Here we denote D ◦ i = D i \ ∂D i .(1) each D i is attached to L cleanly along its boundary, i.e., transversely in thedirections complementary to the tangent lines T ( ∂D i ),(2) D ◦ i ∩ L = ∅ ,(3) D ◦ i ∩ D ◦ j = ∅ , i (cid:54) = j ,(4) the curves ∂D i ⊂ L have minimal pairwise intersections, i.e., there is adiffeomorphism L → T taking each ∂D i to a geodesic of the flat metric.With this definition, the above mentioned configuration( T Cl , { D , D , D } )as drawn in Figure 9 is nothing but an example of Lagrangian seed.The following is the precise statement on which we will be based for this inductiveprocedure starting from (1 , ,
1) to arbitrarily given ( a, b, c ). Lemma 5.2 (Compare with Lemma 4.16 [PT]) . Denote M = C P \ E . Considerthe Clifford torus L and a Lagrangian disk D so that ( L , D ) ⊂ M is a mutationconfiguration. Denote by θ M the Liouville one-form of the exact symplectic form ( ω FS ) | M . Then(1) any neighborhood of L ∪ D contains another mutation configuration ( L , D ) ,(2) there is an arbitrarily small neighborhood U of L ∪ D such that ( θ M ) | U isLiouville, and such that the completion of U is isomorphic to the completionof M . The easiest way to see that we can make another mutation configuration as closeto the given on as we want is to slide all the nodes of an ATF very close to themonotone fibre – see Figure 9. Say that all the nodes are now inside a small disk D in the base. All mutations can then be achieved by sliding the nodes inside D , sothat the fibration remains unchanged in the complement of D . In other words, allthe monotone fibre live in the pre-image of D – see Figure 9.We would like to emphasize that this mutation operation is done in a way that theambient symplectic form, say, the Fubini-Study form unchanged. In particular theunion of all these tori is not dense in C P with respect to the Fubini-Study metric.In fact, our construction shows that the whole family { T a,b,c } can be put into anopen set of arbitrarily small volume by taking the above mentioned neighborhoodas small as we want. Figure 9.
An ATF of C P , with nodes very close to the monotonefibre. All mutations can be made in D , the inside of the dashedcircle, i. e., there are representatives of all T a,b,c tori living inside D . The dashed circle can be infinitesimally small.5.2. Ball packing in the complement of all these tori.
Inside the standardtoric diagram of C P , L Sk projects into the barycentre, union the three segmentsfrom the barycentre to the vertices, illustrated as dashed lines in Figure 10.So, as Figure 10 also illustrates, we can find 9 symplectic balls of the same sizein the complement of L Sk , hence in the complement of all T a,b,c Lagrangian tori, forany capacity smaller than the capacity of the monotone ball.
Figure 10.
For any given capacity c smaller than the capacity ofthe monotone ball, we can find 9 balls in the complement of L Sk ,and hence in the complement of representatives of all T a,b,c tori. XOTIC LAGRANGIAN TORI T a,b,c This is the maximum we can get for volume reasons.6.
Ball packing in the complement of individual torus T a,b,c We can easily see three monotone balls of the same size in the complement of theClifford torus in C P , via the toric blowup.In [CS, Section 7], Chekanov-Schlenk ask if one can embed the Chekanov torusinto the monotone C P C P . The answer is yes and we can indeed easily find twoextra monotone balls in the complement of the Chekanov torus inside C P C P –see Figure 11. Figure 11.
Two extra monotone balls in the complement of theChekanov torus in C P C P . The Chekanov torus projects overthe red segment – see [CS, Section 7]. Figure 12.
Two triangles of the same size in a ATF diagram of C P . The second can be thought to be obtained from the first by“sliding it through the cut”. Note that one edge corresponds to theeigen-direction of the cut, hence it is not distorted as it passesthrough. We recall that in an almost toric fibration, we do have the fibres over the cuts,only the affine structure of the base diagram is not corresponding with the standardaffine structure of R . In particular, we can have a triangular region passing throughthe cut – see Figure 12. The monodromy may distort the edges of the triangle as itcrosses the cut.This allow us to get even better results for the ball packing. Start with aconfiguration of 5 consecutive balls, similar to the ones in Figure 10. Consider anATF with two cuts very close to the monotone Cliford torus. Slide this 5 ballsthrough the cuts as illustrated in Figure 12. You can then “inflate the triangles”, soall of them become monotone triangles. You get a diagram as illustrated in Figure13. Figure 13.
Five monotone balls in the complement of representa-tives of T ,b,c tori.Of course, when we want to embed the monotone balls, we need a tiny neigh-bourhood of the monotone triangle. So all these triangles need to be spaced outby a tiny amount, which is not drawn on Figure 13 for visual purposes. Figure 14illustrates how the balls look like when we undo the monodromies associated withthe cuts for better understanding.These 5 monotone balls are indeed in the complement of tori of the form T ,b,c (1 + b + c = 3 bc ), all together, in particular of both Clifford and Chekanov tori.This is because all these tori are obtained by changing the ATF in the pre-image ofa small region containing the monotone fiber and the two singular fibres – recall theanalogous discussion given in Figure 9. Remark 6.1.
We cannot use this trick in the ATF illustrated in Figure 9, to get amonotone ball in the complement of L Sk and, hence, of all tori T a,b,c simultaneously.If you try to “slide one triangle of Figure 10 through a cut”, with a triangle of sizeclose to the monotone one, it will be forced to cross all the three cuts several times,in a spiral fashion, before eventually entering the dashed neighbourhood in Figure 9. XOTIC LAGRANGIAN TORI T a,b,c Figure 14.
On the top, a configuration of five monotone balls.On the bottom, the same configuration drawn after introducingtwo cuts.To proceed further, we need to make some computations regarding Markov triples( a, b, c ), a + b + c = 3 abc. The following is well-known
Lemma 6.2. If c ≥ a the a (cid:48) = 3 bc − a > c .Proof. c ≥ a ⇒ bc ≥ a ⇐⇒ a (cid:48) = 3 bc − a ≥ bc ⇒ a (cid:48) > c since 2 b ≥ (cid:3) In particular, if a ≤ b ≤ c , then a (cid:48) = 3 bc − a > c and b (cid:48) = 3 ac − b > c . Lemma 6.3 (Section 3.7 of [KN]) . Two out of the three possible mutations of theMarkov triple ( a, b, c ) increase the sum a + b + c and the other reduces it. (In fact, we can indeed deduce from this that if a ≤ b < c then c (cid:48) ≤ b , but wewon’t need that.) Proposition 6.4.
Suppose c > b ≥ a . Then c a + b + c ≥ . (6.1) Proof.
We first transform (6.1) as follows: c a + b + c ≥ ⇐⇒ c ≥ a + b ) ⇐⇒ c ≥ ab − c ) c ⇐⇒ ⇐⇒ ab − c ≤ c ⇐⇒ ab ≤ c. We will prove, by induction on the biggest Markov number, that if c ≥ ab ≤ c . This holds for our base ( a, b, c ) = (1 , , ab ≤ c for a ≤ b < c and consider mutations that increase thebiggest Markov number in the triple. We derive from Propositions 6.2 and 6.3that the mutations that increase the biggest Markov number in the triple are a ↔ a (cid:48) = 3 bc − a and b ↔ b (cid:48) = 3 ac − b being a (cid:48) the biggest number in the triple( b, c, a (cid:48) ) and b (cid:48) the biggest number in the triple ( a, c, b (cid:48) ).So we need to show that 2 bc ≤ a (cid:48) ac ≤ b (cid:48) But we already saw that in the proof of Proposition 6.2: c ≥ a ⇒ bc ≥ a ⇐⇒ a (cid:48) = 3 bc − a ≥ bcc ≥ b ⇒ ac ≥ b ⇐⇒ b (cid:48) = 3 ac − b ≥ ac This finishes the proof. (cid:3)
The inequality (6.1) means that, if the affine lengths of the edges are a , b , c ,then the longest edge c has at least / of the sum of the affine lengths a + b + c .The sum of lengths of the edges is 9 times the size of the base of the monotonetriangle. This means that the longest edge has at least 9 = 6 times the length ofthe base of the monotone triangle. Hence we can see at least 5 monotone balls inthe complement of T a,b,c , for c ≥
2, see Figure 15.Since we have already showed in Figure 13 that we can find 5 monotone ballsin the complement of the Clifford torus T (1 , , T a,b,c for all ( a, b, c ) ∈ M . Figure 15. T a,b,c fibre in an ATF of C P , for c ≥ Theorem 6.5.
Any T a,b,c tori, in particular the Chekanov torus, can be embeddedinto the monotone C P k C P for k ≤ . This affirmatively answers to a question asked by Chekanov and Schlenk [CS,Section 7]. (See Theorem 1.7 and the discussion around it.)
XOTIC LAGRANGIAN TORI T a,b,c In the complement of an elliptic curve
By [Sym2, Proposition 8.2], we know that the preimage E of the edges of a almosttoric fibration diagram, with no rank 0 singularities (i.e., all nodes pushed inside)is a smooth symplectic torus representing the anticanonical divisor. By a resultof Sikorav [Sik2, Theorem 3], see also [ST], we can assume that this boundary isindeed an elliptic curve.In this section we improve the estimate (4.1) further combining the results fromthe previous section.From Proposition 6.4, we see that if c ≥ T a,b,c in an ATF of C P \ E , triangles as close as we need to the triangle of heightequal to the height of the monotone triangle and base equal 6 times the base of themonotone triangle – see Figure 16. Figure 16.
For c ≥ π in the complement of any T a,b,c monotone fibre in an ATF of C P \ E projecting into diamonds converging to ♦ ( / ) (shaded).Inscribed inside one of these triangles we can embed a square of sides with lengthas close to 6 / ♦ (
67 13 ) = ♦ ( ) – see Figure 16. Hence we can embed a ball of capacity as close to π as we want. Figure 17.
Balls of capacities converging to π in the complementof the monotone Clifford torus fibre in an ATF of C P \ E . Moreover, in the complement of the Clifford torus T Cl ∼ = T , , in C P \ E wecan get diamonds as close to ♦ ( ) as we want, hence with capacities converging to π – see Figure 17.We summarize the above discussion into the following Theorem 7.1. inf ( a,b,c ) ∈ M c G ( C P \ E ; T a,b,c ) ≥ π > π Discussion and open questions
In this section, we would like to propose a few interesting open questions inrelation to the geometry of T a,b,c in addition to Problem 1.5.8.1. Hamiltonian-minimal representative.
As mentioned in the introduction,the starting point of our research in the present article lies in our attempt tovisualize the T a,b,c tori in terms of the geometry of Fubini-Study metric on C P .One interesting question is to find a geometric realization of the tori T a,b,c withspecial Riemannian geometric properties in the spirit of [Oh1, Oh2, Oh3]. Question 8.1. (1) Construct a
Hamiltonian-minimal representative in eachHamiltonian isotopy class of T a,b,c with respect to the Fubini-Study metricand visualize the representative.(2) More boldly prove that there exists a Hamiltonian-minimal representativeof T a,b,c in its Hamiltonian isotopy class, or that there is a lower bound ofthe volume inside the Hamiltonian isotopy class.(3) Is there any alternative group theoretic construction of T a,b,c ?According to [Oh3], the mean curvature flow of Lagrangian submanifolds inEinstein-K¨ahler manifolds such as in C P equipped with Fubini-Study metric g preserves the Lagrangian property and decreases the volume.On the other hand, it follows from [Oh2] and the index calculation given byUrbano [Urb] that any Hamiltonian-stable
Hamiltonian-minimal Lagrangian toruswith respect to the Fubini-Study metric is isometric to the Clifford torus in C P , provided it is smooth . Therefore none of smooth representative T a,b,c are Hamiltonian-stable unless ( a, b, c ) = (1 , , smooth volume minimizing representative of T a,b,c in its Hamiltonian isotopy class unless( a, b, c ) = (1 , , Question 8.2.
Consider the set { g · R P | g ∈ Iso( C P ) } , i.e., the set of totallygeodesic R P . Is is true that T a,b,c ∩ ( g · R P ) (cid:54) = ∅ for all g ∈ Iso( C P ). HereIso( C P ) denotes the isometry group of the Fubini-Study metric.Since Iso( C P ) ⊂ Ham( C P ), the above mentioned non-intersection result followsfrom the Floer theoretic question whether or not T a,b,c ∩ φ ( R P ) (cid:54) = ∅ for all φ ∈ Ham( C P ). It turns out that this intersection result depends on the types ofMarkov triples ( a, b, c ). For example, Alston-Amorim [AA] proved that the Cliffordtorus T Cl ∼ = T , , intersects φ ( R P ) for all φ : They proved that a version of Floer XOTIC LAGRANGIAN TORI T a,b,c cohomology between the product T Cl × T Cl and R P × R P in C P × C P is definedand is non-zero even though the Floer cohomology between R P and T Cl is notdefined. On the other hand, the case ( a , b , c ) = (1 , ,
4) i.e., T , , was studied by Wei-Wei Wu [Wu] for which the fiber at the singular vertex of the moment polytope issymplectomorphic to R P and so it does not intersect the semi-toric fiber T (1 , , ∼ = T , , . (It is also shown in [OU] that T , , is the Chekanov torus.) In fact, such anon-intersection result can be proved for any triple ( a, b, c ) one of whose elementis 2. This can be seen by mutating the smooth vertices in Wu’s semi-toric picture[Wu], for instance.These observations lead us to proposing the following conjecture which is aninteresting subject of future investigation. Conjecture 8.3.
There exists a Hamiltonian diffeomorphism φ on C P such that T a,b,c ∩ φ ( R P ) = ∅ if and only if ( a, b, c ) = (2 , b, c ) . Finally, under the assumption that the answer to the second question above isaffirmative, the following question is interesting to ask.
Question 8.4.
Is there a family { T a,b,c } such that there exists a positive constant (cid:15) ∞ > a,b,c ∈ M vol g ( T a,b,c ; C P )( abc ) / ≥ (cid:15) ∞ ?8.2. Size of Weinstein neighborhood of T a,b,c . Another question is related tothe size of the maximal Darboux-Weinstein neighborhood of T a,b,c . We start withsome general discussion on Darboux-Weinstein chart. Let L ⊂ M be a compactLagrangian submanifold. Consider the Darboux-Weinstein chart Φ : U → V where U is a neighborhood of L in M and V is a neighborhood of the zero section o L ⊂ T ∗ L .Then by definition, we have ω = Φ ∗ ω , ω = − dθ for the Liouville one-form θ on T ∗ L and Φ | L = id L under the identification of L with o L .Fix any Riemannian metric g on L . For x ∈ U , we define (cid:107) x (cid:107) g, Φ = (cid:107) Φ( x ) (cid:107) g ( π (Φ( x )) where Φ( x ) ∈ T ∗ π (Φ( x )) L and π : T ∗ L → L is the canonical projection, and (cid:107) · (cid:107) g ( q ) is the norm on T ∗ q L induced by the inner product g ( q ). Definition 8.5.
Let L ⊂ M be a compact Lagrangian submanifold equipped witha metric g . Consider the Darboux-Weinstein chart Φ : U → V . Define w DW (Φ; g ) := inf q ∈ L (cid:32) sup x ∈ π − ( q ) ∩ U (cid:107) x (cid:107) g, Φ (cid:33) and w DW ( L ; M ) = sup Φ w DW (Φ; g ) (8.1)over all Darboux-Weinstein chart of L . We call w DW ( L ; M ) the Weinstein width of L (relative to the metric g ). The w DW ( L ; M ) is another symplectic invariant of L which measures extrinsiccomplexity of the embedding L ⊂ M . Obviously w DW ( L ; M ) > w DW (Φ; g ) > L .With this preparation, we propose the following conjecture which is another wayof examining the conjectural ergodic behavior of the family T a,b,c . Conjecture 8.6.
Consider T a,b,c ⊂ C P . Then inf ( a,b,c ) ∈ M w DW ( T a,b,c ; C P ) = 0 . Proving this conjecture is essentially equivalent to proving the infimum over( a, b, c ) ∈ M of the size of the shape invariant Sh T a,b,c ( C P ) is zero. See [Sik1] forthe definition of the shape invariant and [STV, Section 6] for the relevant study ofthis shape invariants. References [AA] G. Alston and L. Amorim. Floer cohomology of torus fibers and real Lagrangians in Fanotoric manifolds.
Internat. Math. Res. Notices , (12):2751–2793, 2012. 8.1[Abr] M. Abreu. K¨ahler metrics on toric orbifolds.
J. Differential Geom. , 58(1):151–187, 2001.2.2, 2.3, 2.7[Aud] M. Audin.
The topology of torus actions on symplectic manifolds , volume 93 of
Progressin Mathematics . Birkh¨auser Verlag, Basel, 1991. Translated from the French by the author.2.2[BC] P. Biran and O. Cornea. Rigidity and uniruling for Lagrangian submanifolds.
Geom. Topol. ,13:2881–2989, 2009. (document), 1.3, 4[Bir1] P. Biran. Lagrangian barriers and symplectic embeddings.
Geom. Funct. Anal. , 11(3):407–464, 2001. (document)[Bir2] P. Biran. Lagrangian non-intersections.
Geom. Funct. Anal. , 16(2):279–326, 2006. (docu-ment)[CDG] D. M. J. Calderbank, L. David, and P. Gauduchon. The Guillemin formula and K¨ahlermetrics on toric symplectic manifolds.
J. Symplectic Geom. , 1(4):767–784, 2003. 2.12[CO] C.-H. Cho and Y.-G. Oh. Floer cohomology and disc instantons of Lagrangian torus fibersin Fano toric manifolds.
Asian J. Math. , 10(4):773–814, 2006. 2.2, 2.5, 2.5, 2.15[CS] Y. Chekanov and F. Schlenk. Notes on monotone Lagrangian twist tori.
Electron. Res.Announc. Math. Sci. , 17:104–121, 2010. (document), 6, 11, 6[EP] Y. Eliashberg and L. Polterovich. Unknottedness of Lagrangian surfaces in symplectic4-manifolds.
Internat. Math. Res. Notices , (11):295–301, 1993. (document)[ES] J. D. Evans and I. Smith. Markov numbers and Lagrangian cell complexes in the complexprojective plane.
Geom. Topol. , 22(2):1143–1180, 2018. 2.3[Gui] V. Guillemin. Kaehler structures on toric varieties.
J. Differential Geom. , 40(2):285–309,1994. 2.3[Kar] Y. Karshon. Appendix (to D. McDuff and L. Polterovich).
Invent. Math. , 115(1):431–434,1994. 3[Kho] T. Khodorovskiy. Symplectic rational blow-up. arXiv:1303.2581 , 2013. 2.3[KN] B. V. Karpov and D. Y. Nogin. Three-block exceptional sets on del Pezzo surfaces.
Izv.Ross. Akad. Nauk Ser. Mat. , 62(3):3–38, 1998. 6.3[LMS] J. Latschev, D. McDuff, and F. Schlenk. The Gromov width of 4-dimensional tori.
Geom.Topol. , 17(5):2813–2853, 2013. 3.2, 3.2[LS] N. C. Leung and M. Symington. Almost toric symplectic four-manifolds.
J. SymplecticGeom. , 8(2):143–187, 2010. 3.1[LT] E. Lerman and S. Tolman. Hamiltonian torus actions on symplectic orbifolds and toricvarieties.
Trans. Amer. Math. Soc. , 349(10):4201–4230, 1997. 2.3, 2.7[MP] A. Mandini and M. Pabiniak. On the Gromov width of polygon spaces.
Transform. Groups ,23(1):149–183, 2018. (document), 3, 3.2, 3.2[MW] J. Marsden and A. Weinstein. Reduction of symplectic manifolds with symmetry.
Rep.Mathematical Phys. , 5(1):121–130, 1974. 2.3
XOTIC LAGRANGIAN TORI T a,b,c [Oh1] Y.-G. Oh. Second variation and stabilities of minimal Lagrangian submanifolds in K¨ahlermanifolds. Invent. Math. , 101(2):501–519, 1990. 8.1, 8.1[Oh2] Y.-G. Oh. Volume minimization of Lagrangian submanifolds under Hamiltonian deforma-tions.
Math. Z. , 212(2):175–192, 1993. 8.1, 8.1[Oh3] Y.-G. Oh. Mean curvature vector and symplectic topology of Lagrangian submanifolds inEinstein-K¨ahler manifolds.
Math. Z. , 216(3):471–482, 1994. 8.1, 8.1[Oh4] Y.-G. Oh. Addendum to Floer cohomology of Lagrangian intersections and pseudo-holomorphic discs, i.
Comm. on Pure and Applied Math. , 48(11):1299–1302, 1995. (docu-ment)[OU] J. Oakley and M. Usher. On Certain Lagrangian Submanifolds of S × S and C P n . arXiv:1311.5152 , 2013. 8.1[PT] J. Pascaleff and D. Tonkonog. The wall-crossing formula and Lagrangian mutations. arXiv:1711.03209 , 2017. (document), 5.1, 5.1, 5.2[Sch] F. Schlenk. Embedding problems in symplectic geometry , volume 40 of
De Gruyter Exposi-tions in Mathematics . Walter de Gruyter GmbH & Co. KG, Berlin, 2005. 3.2[Sik1] J.-C. Sikorav. Rigidit´e symplectique dans le cotangent de T n . Duke Math. J. , 59(3):759–763,1989. 8.2[Sik2] J.-C. Sikorav. The gluing construction for normally generic J -holomorphic curves. In Sym-plectic and contact topology: interactions and perspectives (Toronto, ON/Montreal, QC,2001) , volume 35 of
Fields Inst. Commun. , pages 175–199. Amer. Math. Soc., Providence,RI, 2003. 7[ST] B. Siebert and G. Tian. On the holomorphicity of genus two Lefschetz fibrations.
Ann. ofMath. (2) , 161(2):959–1020, 2005. 7[STV] E. Shelukhin, D. Tonkonog, and R. Vianna. Geometry of symplectic flux and Lagrangiantorus fibrations. 8.2[STW] V. Shende, D. Treumann, and H. Williams. On the combinatorics of exact Lagrangiansurfaces. arXiv: 1603.07449v1 . (document), 5.1[Sym1] M. Symington. Generalized symplectic rational blowdowns.
Algebr. Geom. Topol. , 1:503–518, 2001. 2, 2.4[Sym2] M. Symington. Four dimensions from two in symplectic topology. In
Topology and geometryof manifolds (Athens, GA, 2001) , volume 71 of
Proc. Sympos. Pure Math. , pages 153–208.Amer. Math. Soc., Providence, RI, 2003. (document), 2.1, 2.1, 2.1, 2.4, 3.1, 7[Ton] D. Tonkonog. From symplectic cohomology to Lagrangian enumerative geometry. arXiv:1711.03292 , 2017. (document), 5.1[Urb] F. Urbano. Index of Lagrangian submanifolds of C P n and the Laplacian of 1-forms. Geom.Dedicata , 48(3):309–318, 1993. 8.1[Via1] R. Vianna. On exotic Lagrangian tori in CP . Geom. Topol. , 18(4):2419–2476, 2014.(document), 2, 2.1, 2.1, 2.1, 2.5[Via2] R. Vianna. Infinitely many exotic monotone Lagrangian tori in CP . J. Topol. , 9(2):535–551,2016. (document), 1.4, 2, 2.3, 2.3, 2.3, 2.3, 2.3, 2.4, 2.4, 2.5[Via3] R. Vianna. Infinitely many monotone Lagrangian tori in del Pezzo surfaces.
Selecta Math.(N.S.) , 23(3):1955–1996, 2017. 3.1[Wit] E. Witten. Phases of N=2 theories in two-dimensions.
Nucl. Phys. , B403:159–222, 1993.[AMS/IP Stud. Adv. Math.1,143(1996)]. 2.2[Wu] W. Wu. On an exotic Lagrangian torus in C P . Compositio Math. , 151(7):1372–1394, 2015.8.1[Zun1] N. T. Zung. A note on focus-focus singularities.
Differential Geom. Appl. , 7(2):123–130,1997. 2.1[Zun2] N. T. Zung. Symplectic topology of integrable Hamiltonian systems. II. Topologicalclassification.
Compositio Math. , 138(2):125–156, 2003. 3.1
Weonmo Lee, Department of Mathematics, POSTECH, Pohang, Korea & Center forGeometry and Physics, Institute for Basic Sciences (IBS), Pohang, Korea
E-mail address : [email protected] Yong-Geun Oh, Center for Geometry and Physics, Institute for Basic Sciences (IBS),Pohang, Korea & Department of Mathematics, POSTECH, Pohang, Korea
E-mail address : [email protected] Renato Vianna, Institute of Mathematics, Federal University of Rio de Janeiro(UFRJ), Rio de Janeiro, Brazil
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