On the Minimal Symplectic Area of Lagrangians

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Then the chord conjecture for subcritically fillable contact manifolds follows from that the minimal symplectic area is bounded from above by the displacement energy of the subcritical domain W by [12]. Our main theorem is finding new families of symplectic manifolds with universally bounded minimal symplectic area, which are not necessarily displaceable. Theorem 1.3. Let W be an symplectically aspherical domain and R be commutative ring. ∗ (1) If SH (W ; R) = 0, then Amin(Lall,W ) < ∞. (2) If W is an exact domain and admits a k-semi-dilation in R coefficient [41], then Amin(LK(π,1),W ) < ∞. ′ ′ ′ Remark 1.4. It is not true that W ⊂ W implies that Amin(L,W ) ≤ Amin(L,W ) since π2(W ,L) → n 2n π2(W, L) may not be surjective. For example, if L = {T }, then Amin(L,B (1)) < ∞. On the other ∗ n n ∗ n hand Amin(L, Dǫ T ) = ∞ as it contains an exact T , but the disk bundle Dǫ T with small enough radius ǫ can be embedded in to the ball B2n(1). However if W ′ ⊂ W is a homotopy equivalence, then we have ′ Amin(L,W ) ≤ Amin(L,W ). In particular, we can use Amin(L,W ) to define a generalized symplectic capacity for star shaped domains in Cn, e.g. the Lagrangian capacity [14, 16]. By [24], if an exact domain W is displaceable, then SH∗(W ; Z) = 0. But the vanishing of symplectic cohomology does not imply being displaceable. Although the vanishing of symplectic cohomology is a restricted condition, there are still many examples (1) flexible Weinstein domains [9, 31], (2) subflexible Weinstein domains [31], (3) V × D for any exact domain V [32], (4) prequantization line bundles over symplectically aspherical manifolds [33]. Moreover, there are many non-flexible examples whose symplectic cohomology vanishes in certain finite field coefficient [4, 25]. k-semi-dilation is a generalization of vanishing of symplectic cohomology, as 0-semi-dilation is equivalent to the vanishing of symplectic cohomology. Examples with k-semi-dilations include cotangent bundles of simply connected manifolds [41, Proposition 5.1], more generally tree plumbings of cotangent bundles of simply connected (spin) manifolds of dimension at least 3 [26, Proposition 6.2], and Milnor fibers of many canonical singularities [41, Theorem A]. Moreover, the k-semi-dilation is preserved under subcritical and flexible surgeries, products and Lefschetz fibrations [41, §3.5]. In this paper, we also find more examples with k-semi-dilations, see Theorem 1.9. Therefore there is a rich class of examples that Theorem 1.3 can be applied. There are also examples of vanishing of symplectic cohomology using local systems [7], as a corollary we have the following.

Theorem 1.5. Let Q be a closed manifold such that the Hurewicz map π2(Q) → H2(Q) is nontrivial. Then ∗ ∗ Amin(L, D Q) is uniformly bounded for any Lagrangian L such that π2(L) → π2(T Q) is trivial. The topological conditions in (2) of Theorem 1.3 and 1.5 is necessary. For example, the zero section of D∗S2 is exact, hence has infinite minimal symplectic area. However, (2) of Theorem 1.3 or 1.5 imply that oriented Lagrangians that are not spheres have a universal minimal symplectic area bound. Indeed, there are many such Lagrangians, as we can take several copies of the zero section after different Hamiltonian perturbations and then use Lagrangian surgeries to resolve the intersections to get smooth Lagrangians with higher genus. Combining Theorem 1.2, 1.3, and 1.5 together, we prove the chord conjecture for the following examples. Corollary 1.6. Let Y be a contact manifold and R a commutative ring. (1) If Y has an exact filling W with SH∗(W ; R) = 0, then the chord conjecture holds for Y . (2) If Y has a symplectically aspherical filling W with SH∗(W ; R) = 0, then the chord conjecture holds for any Legendrian Λ such that the composition π1(Λ) → π1(W ) is trivial, in particular, any simply connected Legendrian. ONTHEMINIMALSYMPLECTICAREAOFLAGRANGIANS 3

(3) If Y has an exact ﬁlling W with a k-semi-dilation in R-coeﬃcient, then the chord conjecture holds for Y and any K(π, 1) Legendrian. ∗ (4) If Y = S Q with the Hurewicz map π2(Q) → H2(Q) nontrivial, then the chord conjecture holds for ∗ Y and any Legendrian Λ such that π2(Y ) → π2(T Q) is trivial.

Remark 1.7. For exact domains W with SH∗(W ) = 0, Ritter showed that the chord conjecture holds for fillable Legendrians via showing the vanishing of the wrapper Floer cohomology of the filling [34]. However, not every Legendrian is fillable and we do not have an effective criterion to prove a Legendrian is fillable.

We first explain the mechanism behind the proof of Theorem 1.3. In the extreme case, if L is an exact Lagrangian inside an exact domain W , in particular, Amin(L,W ) = ∞. In this case, we have the Viterbo transfer map SH∗(W ) → SH∗(T ∗L) preserving various structures. Since SH∗(T ∗L) is never zero and SH∗(T ∗L) does not carry a k-semi-dilation if L is a K(π, 1) space, if SH∗(W ) = 0 then W has no exact Lagrangians, if W admits a k-semi-dilation, then W has no exact K(π, 1) Lagrangians. When L is not an exact Lagrangian, the Viterbo transfer map (without deformation) does not exist. However a truncated version of the Viterbo transfer map still exists, where the threshold of the truncation depends on Amin(L,W ). Then Theorem 1.3 follows from the same argument as the exact case above. Inspired by the argument in [28], it is natural to look for the extreme case of the failure of finite minimal symplectic area in the symplectization, i.e. exact Lagrangians in a symplectization. By [30], there exist abundant exact Lagrangians in overtwisted contact manifolds of dim ≥ 3. Therefore the approach in [28] is not applicable to all contact manifolds. On the other hand, every exact Lagrangian in the symplectization is displaceable by Hamiltonian diffeomorphisms [30, Remark 4]. In particular, there is no exact Lagrangian in the symplectization if the contact manifold is exactly fillable, for otherwise the Lagrangian Floer cohomology is well-defined and non-vanishing, contradicting that it is displaceable. It is an interesting to understand if there is some quantitative shadow of this argument which leads to existence of Reeb chords. Mohnke’s construction was modified by Lisi [27] to obtain existence of homoclinic orbits in the case of displaceable exact domains. Here the displaceable property is again used to obtain universal minimal symplectic area upper bounds by [12]. With new minimal symplectic area bound given by Theorem 1.3, we obtain the following result.

Corollary 1.8. Assume (W, λ) is an exact domain such that SH∗(W ; R) = 0 for a commutative ring R. Let H : W → R be a Hamiltonian with H(x0) = 0 and x0 a hyperbolic zero of the Hamiltonian vector ﬁeld XH . −1 −1 −1 Suppose H (0) is compact and H (0)\{x0} is of restricted contact type, i.e. λ(XH ) > 0 on H (0)\{x0}, then there is an orbit of XH homoclinic to x0.

We also ﬁnd many new examples with k-semi-dilation, which provide more examples for Theorem 1.3 and Corollary 1.6.

Theorem 1.9. Let X be a degree m smooth hypersurface in CPn+1 for m ≤ n. Then for any holomorphic section s of O(k) for k ≤ n + 1 − m, the aﬃne variety X\s−1(0) admits a k + m − 2-semi-dilation.

Acknowledgements. The author would like to thank Laurent Côtéfor explaining the non-existence of exact Lagrangians in the symplectization of an exact fillable contact manifold and pointing out [27, 30]. The author is supported by the National Science Foundation under Grant No. DMS-1926686. It is a great pleasure to acknowledge the Institute for Advanced Study for its warm hospitality. 4 ZHENGYI ZHOU

2. Symplectic cohomology In this section, we review brieﬂy the construction of symplectic cohomology for exact and symplectically aspherical domains and the Viterbo transfer map following [17].

2.1. Symplectic cohomology.

2.1.1. Symplectic cohomology for exact domains. Let (W, λ) be an exact filling and (W , λ)= W ∪∂W ×(1, ∞) be the completion. Let H be a time-dependent Hamiltonian on the completion (W , λ), then the symplectic c b action for an orbit γ is c b ∗ AH (γ)= − γ λ + H ◦ γdt, (2.1) Z ZS1 b where dλ(·, XH )=dH. We say an almost complex structure J is cylindrical convex near ∂W × {r0} iff near r = r we have that J is compatible with dλ, J preserves ξ := ker λ| and J(r∂r)= R , where R is 0 b ∂W ×{r} λ λ the Reeb vector field on ∂W ×{r}. We will consider H, which is a C2 small perturbation of the Hamiltonian b b that is 0 on W and linear with slope a on ∂W × (1, ∞), such that a is not a period of Reeb orbits of Rλ on ∂W × {1}. We may assume the Hamiltonian is non-degenerate, then the periodic orbits consist of the following.

(i) Constant orbits on W with AH ≈ 0. (ii) Non-constant orbits near Reeb orbits of Rλ on ∂W × {1}, with action close to the negative period of the Reeb orbits, in particular, AH ∈ (−a, 0). After ﬁxing a compatible almost complex structure that is cylindrical convex near a slice where the Hamil- tonian is linear with slope a, we can consider the compactiﬁed moduli space of Floer cylinders, i.e. solutions to ∂su + J(∂tu − XH ) = 0 modulo the R translation and asymptotic to two Hamiltonian orbits

R 1 R Mx,y = u : s × St → W ∂su + J(∂tu − XH ) = 0, lim u(s, ·)= x, lim u(s, ·)= y / . s→∞ s→−∞

c The count of rigid Floer cylinders defines a cochain complex C(H), which is a free R module generated by Hamiltonian orbits, for any commutative ring R. The differential δ0 is defined as

0 δ (x)= #Mx,yy.

y,dimXMx,y=0 ∗ The orbits of type (i) form a subcomplex C0(H), whose cohomology is H (W ). The orbits of type (ii) form a quotient complex C+(H). The cochain complexes are graded by n − µCZ , which is in general a Z2 grading unless c1(W ) = 0. Given two Hamiltonians Ha,Hb with slopes a < b, we can consider a non-increasing homotopy of Hamiltonians Hs from Hb to Ha, i.e. Hs = Hb for s ≪ 0 and Hs = Ha for s ≫ 0. Then rigid solution to ∂su + J(∂tu − XHs ) = 0 defines a continuation map C(Ha) → C(Hb), which is compatible with splitting into zero and positive complexes. Then the (positive) symplectic cohomology of W is defined as ∗ ∗ ∗ ∗ SH := lim H (C(Ha)),SH (W ) := lim H (C+(Ha)), a→∞ + a→∞ which fits into a tautological exact sequence, ∗ ∗ ∗ ∗+1 . . . → H (W ) → SH (W ) → SH+(W ) → H (W ) → . . . ONTHEMINIMALSYMPLECTICAREAOFLAGRANGIANS 5

∗ ∗ ∗ ∗ We deﬁne the ﬁltered symplectic cohomology SH

2.2. The Viterbo transfer map. Let (V, λV ) ⊂ (W, ΛW ) be an exact subdomain, i.e. λW |V = λV , then there exists ǫ> 0, such that Vǫ := V ∪ ∂V × (1, 1+ ǫ] ⊂ W . Then we can consider a Hamiltonian H on W as a C2-small non-degenerate perturbation to the following. c (1) H is 0 on V . (2) H is linear with slope B on ∂V × [1, 1+ ǫ], such that B is not the period of a Reeb orbit on ∂V . (3) H is Bǫ on W \Vǫ. (4) H is linear with slope A

Then there are ﬁve classes of periodic orbits of XH . (I) Constant orbits on V with AH ≈ 0. (II) Non-constant periodic orbits near ∂V with AH ∈ (−B, 0). (III) Non-constant periodic orbits near ∂V × {1+ ǫ} with action AH ∈ (Bǫ,Bǫ +(1+ ǫ)B). (IV) Constant orbits on W \Vǫ with AH ≈ Bǫ. (V) Non-constant periodic orbits near ∂W with AH ∈ (Bǫ − A,Bǫ)

In particular, orbits of the form (I), (II) form a quotient complex when Bǫ>A. If we use HV to denote the Hamiltonian on V which is the linear extension of the truncation of H on Vǫ. Then by [17, Lemma 2.2], the ∗ quotient complex is identified with C (HV ) (this is where the exactness of cobordism is crucial). Next we b 2 consider a Hamiltonian HW on W which is a C small perturbation to the function that is zero on W and linear with slope A on ∂W × (1, ∞). Then H ≤ H and we can find non-increasing homotopy from H to c W HW , which defines a continuation map. Therefore we have a map ∗ ∗ ∗ C (HW ) → C (H) → C (HV ).

Since the continuation map increases the symplectic action, the above map respects the splitting into C0,C+. Taking direct limit for B yields the Viterbo transfer map which is compatible with the tautological exact

2The splitting, in particular, the positive symplectic cohomology is still deﬁned by the asymptotic behaviour lemma [17, Lemma 2.3]. 6 ZHENGYI ZHOU sequence,

/ ∗ / ∗ / ∗ / ∗+1 / . . . H (W ) SH

/ ∗ / ∗ / ∗ / ∗+1 / . . . H (V ) SH (V ) SH+(V ) H (V ) . . .

If we also take the direct limit for A, then we get the Viterbo transfer map for the full symplectic cohomology. Moreover, SH∗(W ) is a unital ring by the pair-of-pants construction. And the continuation map H∗(W ) → SH∗(W ) is a unital ring map. If we take SH[0],∗(W ), then the Viterbo transfer map maps to the part of SH∗(V ) generated by orbits that are contractible in W , which can then be included into the full symplectic cohomology generated by all orbits.

2.3. S1-equivariant symplectic cohomology. The S1 action on the free loop space gives rise to an S1- equivariant analogue for symplectic cohomology [11, 36]. More precisely, one can consider Hamiltonians with slope a that are parameterized over ES1 = S∞ (in the actual construction, we need to approximate ES1 by S2m+1). In particular, by counting parameterized Floer trajectories, we have maps δ1, δ2,... defined on the i j S1 ∞ k k regular symplectic cochain complex, such that i+j=k δ ◦δ = 0 for k ≥ 0. Then δ = k=0 u δ defines a −1 −1 ∗ differential on (C(H) ⊗ R[u, u ]) ⊗R[u] (R[u, uP ]/[u]), whose cohomology is denoted byPSHS1,

∗ ∗ ∗ δ ∗+1 . . . → HS1 (W ) → SHS1 (W ) → SH+,S1 (W ) → HS1 (W ) → ...,

∗ 1 ∗ ∗ −1 where each cohomology is equipped with a H (BS )= R[u]-module and HS1 (W ) = (H (W )⊗R[u, u ])⊗R[u] (R[u, u−1]/[u]), i.e. Laurent series with coeﬃcient in H∗(W ) without monomials of positive degree.

∗ 0 Deﬁnition 2.1. Let π0 denote the projection HS1 (W ) → H (W ), we say W carries a k-semi-dilation iﬀ ∗ k+1 there exists x ∈ SH+,S1 (W ), such that π0 ◦ δ(x) = 1 and u (x) = 0.

[0],∗ It is clear from deﬁnition, that k-semi-dilation only depends on the part SH+,S1 (W ) generated by contractible orbits.

Remark 2.2. A symplectic dilation was introduced Seidel-Solomon [37] as an element x ∈ SH1(W ), such that ∆(x) = 1, where ∆ is the BV operator. In [41], we introduce the concept of k-dilation, an exact domain ∗ k+1 W admits a k-dilation iﬀ there exists x ∈ SH+,S1 (W ), such that δ(x) = 1 and u (x) = 0. Then the existence of a 1-dilation is equivalent to the existence of a symplectic dilation, and the existence of a k- dilation is equivalent to the existence of a cyclic dilation for h = 1 introduced in [26].A k-semi-dilation ∗ k+1 was originally introduced in [41] as an element x ∈ SH+,S1 (W ), such that δ(x)|∂W = 1 and u (x) = 0. Deﬁnition 2.1 here imposes weaker condition compared to [41] and enjoys same properties as in [41]. ONTHEMINIMALSYMPLECTICAREAOFLAGRANGIANS 7

Let V ⊂ W be an exact subdomain, by the family version of the construction in §2.2, we have the following Viterbo transfer map which respects the R[u]-module structure, see also [20].

/ ∗ / ∗ / ∗ / ∗+1 / . . . HS1 (W ) SHS1,

/ ∗ / ∗ / ∗ / ∗+1 / . . . HS1 (V ) SHS1 (V ) SH+,S1 (V ) HS1 (V ) . . . In particular, if W carries a k-semi-dilation, then V also carries a k semi-dilation. Example 2.3. The following examples has k-semi-dilation for some k. ∗ ∗ (1) T Q for simply connected manifold Q [41, Proposition 5.1] and more generally plumbings of {T Qi} with respect to any tree for simply connected spin manifold Qi such that dim Qi ≥ 3 [26, Proposition 6.2]; n n (2) The Brieskorn variety x0 + . . . + xn = 1 [41, Theorem A]; (3) Lefschetz fibration whose page admits a k-semi-dilation [41, Proposition 3.27]; (4) Products of exact domains if one of them admits a k-semi-dilation, [41, Proposition 3,26]; (5) Flexible surgery does not affect the existence of k-semi-dilation [41, Proposition 3.28]. Using the functorial property of k-semi-dilation and Lefschetz fibration, there are many more Brieskorn varieties with k-semi-dilation, when the Brieskorn singularity is canonical, see [41, §5]. Example 2.4. Let L be a smooth K(π, 1) space, then T ∗L does not admit a k-semi-dilation. This is because ∗ SHS1 (T L) is the equivariant homology of the free loop space possibly twisted by a local system [2, 3, 35, 39], ∗ ∗ ∗ and SH+,S1 (T L) is generated by non-contractible loops, hence T L can not support a k-semi-dilation for any k.

3. Truncated Viterbo transfer Let L ⊂ W be a Lagrangian. We say L is exact if λ is exact on L. In this case, the complement of a Weinstein neighborhood of L is an exact cobordism. In general, λ|L is only a closed class, and the complement is a strong cobordism and the Viterbo transfer in §2.2 fails. In this case, we have

∗ ∗ Amin(L,W ) = inf γ λ γ ∈ π1(L) that is contractible in W, γ λ> 0 . Z Z

Proposition 3.1. Let (W, λ) be an exact domain and L a Lagrangian such that Amin(L,W ) ≥ 5A> 0, then ∗ ∗ ∗ we have a Viterbo transfer SH

/ ∗ / [0],∗ / [0],∗ / ∗+1 / . . . H (W ) SH

/ ∗ ∗ / ∗ ∗ / ∗ ∗ / ∗+1 ∗ / . . . H (T L) SH (T L) SH+(T L) H (T L) . . .

∗ ∗ Proof. Fixing a generic metric on L, we have the unit cotangent bundle (D L, λstd). We deﬁne (Dǫ L, λstd)= ∗ ∗ ∗ D L\(∂D L×(ǫ, 1]). By the Weinstein neighborhood theorem, we have a symplectic embedding (Dǫ L, dλstd) 1 ∗ 1 into W . WLOG, we can assume there exits a closed one form β ∈ Ω (Dǫ L), which is a pullback from Ω (L), 8 ZHENGYI ZHOU

∗ such that λ = λstd + β on Dǫ L. Then we consider Hamiltonians H on W in the definition of the Viterbo transfer that are C2 small perturbations to functions with the following properties, c ∗ (1) 0 on D ǫ L, 2 2A ∗ ∗ 3 (2) linear with slope with respect to r ∈ (0,ǫ] on Dǫ L\D ǫ L, ǫ 2 ∗ (3) A on W \Dǫ L, (4) linear with slope A on ∂W × (1, ∞). The symplectic action of orbits of type (I), (IV), (V) is the same as before. The symplectic action using λstd for type (II), (III) is the same as before. But the actual symplectic action using λ for type (II), (III) ∗ has a difference from γ β for the orbit γ. Since Γ := Amin(L,W ) ≥ 5A. We known that the symplectic action of contractibleR (in W ) orbits of type (II) is in (−A, 0) + ZΓ and type (III) is in (A, 3A)+ ZΓ. Then we consider the Hamiltonian HW on W , which is admissible with slope A. Since the symplectic action of orbits of X is in (−A, 0) and the continuation induced from the non-increasing homotopy from H to H HW c W increases symplectic action, then by composing with the quotient map to the quotient complex of C[0](H) [0] generated by orbits with action at most 0, we get a cochain map from C (HW ) to the complex generated by orbits of XH that are contractible in W with action in (−A, 0). Since Γ ≥ 5A, no type (III) orbits are in this action window and those of type (II) in the action window have the property that γ∗β = 0. We use ˜ ˜ ∗ ∗ C∗ to denote this complex. We claim C∗ is isomorphic to Cβ(HDǫ L), i.e. the subcomplex generatedR by orbits ∗ ∞ 1 ∗ annihilated by β, where HDǫ L ∈ C (S ×Dǫ L) is the linear extension of the truncation. Since the generators are obviously identified, it suffices to show the differential of C˜∗ is defined by counting holomorphic curves ∗ ˜ d contained in Dǫ L. Let x,y ∈ C∗, since we can pick the almost complex structure to be cylindrical convex for ∗ ∗ ∗ ∗ λstd on Dǫ L\D ǫ L, we claim if u ∈ Mx,y then im u ⊂ D 3 ǫL. Assume otherwise that u escapes D 3 ǫL, and 2 4 4 −1 ∗ ∗ S = u (W \D 3 L). WLOG, we may assume u ⋔ ∂D 3 L, hence S is a compact surface with boundary, then 4 ǫ 4 ǫ ∗ ∗ by homologyc reasons u|∂S represents a class annihilated by β, in particular, ∂S u λ = ∂S u λstd, this allows to apply the integrated maximum principle by Abouzaid-Seidel [5], or [17,R Lemma 2.2].R More precisely, we have

1 2 0 ≤ E(u|S ) := |du − XH ⊗ dt| d vol 2 ZS = u∗dλ − u∗dH ∧ dt ZS = d(u∗λ − u∗Hdt) ZS = u∗λ − (u∗H)dt Z∂S ∗ ∗ = u λstd − (u H)dt Z∂S

≤ λstd(du − XH (u) ⊗ dt) Z∂S

∗ ∗ 3 D ǫ L D L A If we rescale 2 back to , the slope is . ONTHEMINIMALSYMPLECTICAREAOFLAGRANGIANS 9

= λstd(J ◦ (du − XH (u) ◦ (−j)) Z∂S = dr ◦ du ◦ (−j) ≤ 0 Z∂S

˜ ∗ ∗ which is a contradiction. Hence C∗ is isomorphic to Cβ(HDǫ L). It is clear by action reasons that the map ∗ ∗ C(HW ) → Cβ(HDǫ L) respects the splitting into C0,C+. Then the claim follows from the composition with continuation map to the full symplectic cohomology of T ∗L. By the exactly same argument, we have the following parameterized version for equivariant symplectic cohomology.

Proposition 3.2. Let W be an exact domain and L a Lagrangian such that Amin(L,W ) ≥ 5A > 0, then [0],∗ ∗ ∗ we have a Viterbo transfer R[u]-map SHS1,

/ ∗ ∗ / ∗ ∗ / ∗ ∗ / ∗+1 ∗ / . . . HS1 (T L) SHS1 (T L) SH+,S1 (T L) HS1 (T L) . . .

Next, we consider the case for symplectically aspherical case. Let HW be an admissible Hamiltonian on W with slope a, the symplectic action of a contractible orbit γ is given by (2.1) if γ is contractible in ∂W . If γ is only contractible in W , then (2.1), (2.2) might diﬀer. Nevertheless, there exists a function f(a) ≥ a> 0, such that symplectic action of periodic orbits of XHW is supported in (−f(a),f(a)).

Proposition 3.3. Let W be a symplectically aspherical domain and L a Lagrangian such that Amin(L,W ) ≥ [0],∗ ∗ ∗ 10f(A) > 0, then we have a Viterbo transfer SH

∗ ι / [0],∗ H (W ) SH

H∗(T ∗L) / SH∗(T ∗L)

∗ Proof. By the Weinstein neighborhood theorem, we have a symplectic embedding (Dǫ L, dλstd) into W as before. Then we consider Hamiltonians H on W in the definition of the Viterbo transfer that are C2 perturbations to functions with the following properties, ∗ c (1) 0 on D ǫ L, 2 4f(A) ∗ ∗ (2) linear with slope with respect to r ∈ (0,ǫ] on Dǫ L\D ǫ L. ǫ 2 ∗ (3) 2f(A) on W \Dǫ L, (4) linear with slope A on ∂W × (1, ∞). The symplectic action of orbits of type (I), (IV) is the same as before. The symplectic action of orbits of type (V) have action in (f(A), 3f(A)). The actual symplectic action using ω for a contractible orbit γ of ∗ type (II), (III) is differed from the symplectic action using λstd by u ω, where u ∈ π2(W, L) with boundary ∗ homotopic to γ in Dǫ L, this follows from λstd|L = 0. Since Γ := Amin(L,W ) ≥ 10f(A), we known that the 10 ZHENGYI ZHOU symplectic action of type (II) orbit is in (−2f(A), 0) + ZΓ and the symplectic action of type (III) orbits is in (2f(A), 6f(A)) + ZΓ. Then we consider the Hamiltonian HW on W , which is admissible with slope A. Since the symplectic action of orbits of X is in (−f(A),f(A)) and the continuation map C[0],∗(H ) → C[0],∗(H) HW c W increases symplectic action, then by composing with the quotient map to the quotient complex of C[0],∗(H) [0],∗ generated by orbits with action at most f(A), we get a cochain map from C (HW ) to the complex generated by orbits of XH that are contractible in W with action in (−f(A),f(A)). Since Γ ≥ 10f(A), no type (III) orbits are in this action window and those of type (II) in the window have the property that ∗ u ω = 0, where u is the disk as before. We use C˜∗ to denote this complex. We claim that C˜∗ is isomorphic ∗ ∗ toR Cω(HDǫ L), i.e. the subcomplex generated by orbits in the homotopy classes that are contractible in W and ∗ ∗ ∞ 1 ∗ u ω = 0 for the contracting disk u, where HDǫ L ∈ C (S × Dǫ L) is the linear extension of the truncation. SinceR the generators are obviously identified, it suffices to show the differential of C˜∗ is defined by counting ∗ ˜ d holomorphic curves contained in Dǫ L. Let x,y ∈ C∗, since we can pick the almost complex structure to ∗ ∗ ∗ be cylindrical convex for λstd on Dǫ L\D ǫ L, we claim if u ∈ Mx,y then im u ⊂ D 3 L. Assume otherwise 2 4 ǫ ∗ −1 ∗ that u escapes D 3 L, and S = u (W \D 3 L). In view of the argument before, it is sufficient to prove 4 ǫ 4 ǫ ∗ ∗ ∗ ∗ ∗ ∗ S u ω = ∂S λsty. This follows fromc that x,y ∈ Cω(HDǫ L) and we have R×S1 u ω = x λstd − y λstd. RThen theR claim follows from the composition with the continuation map toR the full symplecticR cohomologyR of T ∗L and the standard homotopy argument for continuation maps.

Proof of Theorem 1.3. If W is a symplectically aspherical domain such that SH∗(W ) = 0, then we have

[0],∗ 0 A := min a ι(1) = 0 ∈ SH

∗ ∗ Then we have Amin(L,W ) ≤ 10f(A). Assume otherwise, by Proposition 3.3, we have 1 = 0 in SH (T L), contradicting that SH∗(T ∗L) 6= 0. If W is exact and admits a k-semi-dilation, then we have

∗ π0◦δ 0 A := min a 1 ∈ im(SH+,S1,

∗ If there is a K(π, 1) Lagrangian L ⊂ W with Amin(L,W ) > 5A, then by Proposition 3.2, we have T L admits a k-semi-dilation, contradicting Example 2.4.

Proof of Theorem 1.5. By [7] if π2(Q) → H2(Q) is nontrivial, then there exists a local system ρ on L0Q (the space of contractible loops), such that SH∗(T ∗Q; ρ) = 0. Proposition 3.1 holds for symplectic coho- ∗ mology with local system. Since such local system is classiﬁed by Hominv(π2(W ), C ), i.e. the subset of ∗ ∗ Hom(π2(W ), C ) that is invariant under the π1(W ) conjugation. Since π2(L) → π2(T Q) is trivial, we have ∗ ∗ ∗ ∗ ∗ ∗ C the local system is trivial on Dǫ L, hence SH (Dǫ L; ρ|Dǫ L)= SH (T L; ) 6= 0. Then the proof of Theorem 1.3 can be applied to ﬁnish the proof.

Remark 3.4. The coeﬃcient in Proposition 3.1 is not optimal. With a more careful argument, one should be able to reduce the condition to Amin(L,W ) ≥ A. Then Theorem 1.3 gives an upper bound for the Lagrangian ∗ (torus) capacity by the spectral invariant of that element in SH+(W ) that eliminates 1. However, for the case of ellipsoid, this upper bound is close to optimal iﬀ the ellipsoid is thin and is sharp for cylinders. For a round ball, the Lagrangian capacity is computed in [16]. ONTHEMINIMALSYMPLECTICAREAOFLAGRANGIANS 11

4. The chord conjecture For the completeness, we recall the proof of Theorem 1.2 from [28]. Mohnke’s proof of Theorem 1.2. Let α be a contact form on Y . We may assume there is an exact subdomain W ′ ⊂ (W, λ), such that the Reeb dynamics of λ on ∂W ′ is the same as α up to rescale and W ′ ⊂ W is an ′ homotopy equivalence. Therefore we have Amin(Lall,W ) ≤Amin(Lall,W ) < ∞. Let φt denote the ﬂow on ′ ′ ∂W . If the Legendrian Λ has no Reeb chord, then Φ : Λ × [ǫ, 1] × R → ∂W × [ǫ, 1], (x,s,t) 7→ (φt(x),s) is an embedding. Then it is direct to check that any embedded loop γ ∈ [ǫ, 1] × R gives rise to a Lagrangian ′ ∗ Φ(Λ×γ) ⊂ ∂W ×[ǫ, 1]. Note that λ|Λ =0andΦ (d(rλ))=ds∧dt, then we know that Amin(Φ(Λ×γ),W ) is the the area enclosed by γ, which can be arbitrarily big, contradiction!. The situation for K(π, 1) Legendrians is similar as the product of a K(π, 1) space with S1 is again a K(π, 1) space. ∗ Proof of Corollary 1.6. Case (1), (3) follow Theorem 1.2 and 1.3. For case (4), if π2(Λ) → π2(T Q) is trivial, ∗ then π2(Φ(Λ × γ)) → π2(T Q) is also trivial. Then the existence of Reeb chords follows from Theorem 1.5 and the same argument of Theorem 1.2. In the case (2), if π1(Λ) → π1(W ) is trivial, then we know that 4 Amin(Φ(Λ × γ),W ) is again area enclosed by γ . Then the claim follows from the argument of Theorem 1.2 and Theorem 1.3. Proof of Corollary 1.8. The proof follows from the same argument as [27, Theorem 1.1], where we replace [12] by Theorem 1.3 in the proof of [27, Theorem 2.1] to obtain a universal upper bound of the minimal symplectic area. Remark 4.1. In the special case of SH∗(W ) = 0, the chord conjecture for Legendrian spheres can also be proven by surgery following the idea in [13]. Let W ′ be the domain obtained from attaching a Weinstein handle to the Legendrian sphere Λ. If Λ has no Reeb chords, then by [13], we have the Viterbo transfer map ∗ ′ ∗ SH+(W ) → SH+(W ) is an isomorphism. Note that we have the following commutative diagram, / ∗ ′ / ∗ ′ / ∗ ′ / ∗+1 ′ / . . . H (W ) SH (W ) SH+(W ) H (W ) . . .

/ ∗ / ∗ / ∗ / ∗+1 / . . . H (W ) SH (W ) SH+(W ) H (W ) . . . ∗ ∗ ∗+1 ∗ ′ ∗+1 ′ Since SH (W ) = 0, we have SH+(W ) → H (W ) hits 1. Therefore SH+(W ) → H (W ) also hits 1, ∗ ′ ∗ ∗+1 ∗ because SH+(W ) → SH+(W ) is an isomorphism. As a consequence, we have H (W ) ≃ SH+(W ) ≃ ∗ ′ ∗+1 ′ ∗ ∗ ′ SH+(W ) ≃ H (W ). However, we can not have H (W ) ≃ H (W ), which is a contradiction. Following the philosophy of SFT, if V ⊂ W is not an exact subdomain, i.e. the cobordism W \V is not exact, the strong cobordism W \V gives rise a Maurer-Cartan element by counting holomorphic caps in W \V [15], and the Viterbo transfer map should hold for the symplectic cohomology of V deformed by the Maurer-Cartan element. Then the classical Viterbo transfer map should hold if we truncated at an action ∗ smaller than the minimal action of the Maurer-Cartan element. In the case of V = Dǫ L, the action of a Maurer-Cartan element is the period of a geodesic plus the correction from multiplies of Amin(L,W ). In particular, it is not obvious the Maurer-Cartan action is greater than Amin(L,W ). However, if we let ǫ → 0, we can replace the symplectic cohomology of T ∗L by the string topology of L following [17]. In this case, the Maurer-Cartan element is represented by the evaluation of boundaries of holomorphic disks, whose action is

4 In general, it might be smaller contributed by a loop with a nontrivial π1(Λ) component that is contractible only in W . 12 ZHENGYI ZHOU indeed bounded below by Amin(L,W ). Those two constructions (deformed symplectic cohomology, deformed string topology) faces various technical diﬃculties, whose full constructions with complete analytical details are not available in literature. The argument in this paper can be viewed as a substitute of such idea in the classical construction. The closely related Weinstein conjecture asserts the existence of Reeb orbits on every compact contact manifold. It was proven in dimension 3 by Taubes [38] and various other cases [1, 6, 22]. All the the proofs are based on existence of holomorphic curves, often abundances of holomorphic curves5. The relation between holomorphic curves and Reeb orbits are much better understood than the following question. Question 4.2. How abundances of holomorphic curves can be used to show the existence of Reeb chords. Note that the conditions in Corollary 1.6 imply that the symplectic manifold is uniruled [41, Theorem 3.23].

5. New examples with k-semi-dilation Let X be a degree m smooth hypersurface in CPn+1 for m ≤ n. By [29, Lemma 7.1], for any holomorphic section s of O(k), we have the affine variety X\s−1(0) embeds exactly into X\S−1(0), where S is a holomorphic section of O(k) such that S−1(0) is a smooth divisor with multiplicity 1. In our case, we can assume S−1(0) in the intersection of X with a generic smooth degree k hypersurface in CPn+1. In view of the functorial property of k-semi-dilations, Theorem 1.9 is implied by the following. Theorem 5.1. If k ≤ n + 1 − m, then X\S−1(0) admits a k + m − 2-dilation. To prove this theorem, we will adopt the method in [18] as in [41]. We first recall the setups. We use D to denote the smooth divisor S−1(0) and Xˇ to denote the exact (Weinstein) domain X\D. Then the contact boundary Y := ∂Xˇ is equipped with a Boothby-Wang contact structure such that the Reeb flow is the S1 N ˇ action on the circle bundle with periods 2π · . Then we pick two Morse functions fD on D and fXˇ on X, ˇ such that ∂rfXˇ > 0 near ∂X = Y . We use Crit(f) to denote the set of critical points. We may assume fD is perfect, this follows from h-cobordism when n ≥ 4 and direct check when n = 2. When n = 3, we only need to consider m = 1, k = 1, 2, 3, m = 1, k = 1, 2 and m = 3, k = 1, the existence of perfect Morse function follows from [21] except for the case m = k = 2. When m = k = 2, D is a K3 surface, hence there exists a perfect Morse function on D. We also assume fXˇ has a unique minimum m. In the following, we will consider the filtered cochain complex for period up to 2πk. One useful fact is that the circle with wrapping number around D from 1,...,k are in different homotopy classes of Xˇ. Note that k ≤ n + 1 − m is exactly the monotonicity condition in [18], we list the properties of the symplectic cochain complex and its S1-equivariant analogue from [18] as follows.

(1) The symplectic cochain complex up to period 2kπ is generated by x ∈ Crit(fXˇ ) andp ˇi, pî for p ∈ Crit(fD) and 1 ≤ i ≤ k. The positive cochain complex is generated byp ˇi, pî for p ∈ Crit(fD) and 1 ≤ i ≤ k. 0 s u u (2) We have h δ (ˇpi), qî i = h c1(O(k), [WD(q)] ∩ [WD(p)]), where WD(p) = {q ∈ D| limt→∞ φt(q) = p} with φt is the gradient flow of fD. u ˇ (3) For x ∈ Crit(fXˇ ) such that WXˇ (x) represents a homology class in H∗(X), we have 0 X,D u u h δ (ˇpk),x i = kGW0,1,(k),A([WXˇ (x)], [WD(p)])

5In the context of ECH, the non-triviality of the U-map implies the abundance of holomorphic curves. ONTHEMINIMALSYMPLECTICAREAOFLAGRANGIANS 13

X,D u Here GW0,1,(k),A([m], [WD(p)]) stands for the relative Gromov-Witten invariant counting rational curves in X of the positive generator6 A ∈ H2(X) with two marked points subject to the point u constraint m and the unstable manifold WD(p) of p in D with intersection multiplicity with D is at u 0 least k. If WXˇ (x) is not closed, then there are still possible nontrivial h δ (ˇpk),x i. Since we only 0 care about h δ (ˇpk),m i for the unique minimum m, the formula above suffices for our purpose. (4) There are also Morse differentials between critical points of fXˇ , the above three cases are all the 0 nontrivial terms of δ . This because fD is perfect and p1,...,pk are in different homology classes in Xˇ. 1 i 1 (5) We have δ (ˇpi) =p î and δ = 0 for i ≥ 2. This follows from the S -equivariant transversality augment in [41], where the S1-equivariant transversality is provided by [18]. X,D n+1−m−k CPn+1 Proof of Theorem 5.1. By [19], we have GW0,1,(k),A([pt], [D] ∩ [H]) 6= 0 where H ∈ H2n( ) u is the hyperplane class. We assume p is critical point of fD of index 2(m + k − 2), such that [WD(p)] = n+1−m−k 7 i u i n+1−m−k+i [D]∩ [H] . Similarly we use p to represent the critical point such that [WD(p )] = [D]∩ [H] m+k−2 i i 1 for 1 ≤ i ≤ m+k −2. Then pk + i=1 (−ku) pk represents a closed cochain in the S -equivariant cochain m+k−2 i i X,D n+1−m−k ˇ complex, and π ◦ δS1 (pk + i=1P (−ku) pk) = kGW0,1,(k),A([pt], [D] ∩ [H]) 6= 0. Therefore X carries a m + k − 2-dilation.P Remark 5.2. Following the fibration argument in [41, §5], we can prove that the order of semi-dilation for Xˇ is exactly k + m − 2. It is natural to expect that examples in Theorem 1.9 carry k + m − 2-dilation instead of k + m − 2-semi-dilation. However, this requires checking the vanishing of many other relative Gromov-Witten invariants like [41, Theorem A]. Remark 5.3. One can derive a formula of the positive S1-equivariant symplectic cohomology of smooth divisor complements similar to [18]. Under the monotonicity assumption, we can use S1-equivariant transversality to use a quotient construction of the positive S1-equivariant symplectic cohomology instead of the Borel construction used in this paper, see [11]. In this case, the cochain complex is generated by pk for p ∈ Crit(fD) and k ≥ 1, with differential purely determined by Morse differential of fD. All the other contributions from check orbits to hat orbits in [18, Theorem 9.1] now contributes to the R[u]-module structure. Note that by the Gysin exact sequence [10], the positive S1-equivariant symplectic cohomology as a R[u]-module can recover the regular positive symplectic cohomology. In particular, this is just a reformulation of [18, Theorem 9.1]. When the monotonicity assumption fails, we may not have a simple collections of degeneration as in [18], but the other consequence of monotonicity, namely S1-equivariant transversality, is available through polyfold techniques [43]. In particular, it seems that positive S1-equivariant symplectic cohomology (just as a group not a R[u]-module) is much easier to compute.

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