Computation of annular capacity by Hamiltonian Floer theory of non-contractible periodic trajectories

The first author introduced a relative symplectic capacity $C$ for a symplectic manifold $(N,\omega_N)$ and its subset $X$ which measures the existence of non-contractible periodic trajectories of Hamiltonian isotopies on the product of $N$ with the annulus $A_R=(R,R)\times\mathbb{R}/\mathbb{Z}$. In the present paper, we give an exact computation of the capacity $C$ of the $2n$-torus $\mathbb{T}^{2n}$ relative to a Lagrangian submanifold $\mathbb{T}^n$ which implies the existence of non-contractible Hamiltonian periodic trajectories on $A_R\times\mathbb{T}^{2n}$. Moreover, we give a lower bound on the number of such trajectories.

is Theorem 1.1. We consider (N, ω N ) = (T 2n , ω std ) = (R/2Z × R/Z) n , ω std and T n stands for ({0} × R/Z) n where ω std is the standard symplectic form on T 2n . Theorem 1.1. Let R > 0 and u be real numbers such that u ∈ (−R, R). For any smooth Hamiltonian H : [0, 1] × A R × T 2n → R with compact support and any ∈ Z such that max{R| | + u , 0} ≤ c = inf [0,1]×Lu×T n H, there exists a Hamiltonian periodic trajectory x in the homotopy class (α , 0 T 2n ) ∈ [S 1 , A R × T 2n ] with action A H (x) ≥ c − u , where A H is the action functional defined in Section 2. Moreover, if H is non-degenerate, then the number of such x's is at least The result in Theorem 1.1 is sharp in the sense that for any > 0 there exists a Hamiltonian H : [0, 1]×A R ×T 2n → R with inf [0,1]×Lu×T n H = max{R| |+u , 0}− without periodic trajectory in (α , 0 T 2n ) (see the proof of Theorem 1.2 in Subsection 4.2).
To obtain Theorem 1.1, we will prove the non-zeroness of the homomorphism T [a,∞);c α which is defined in Subsection 2.5. To prove the non-zeroness, we use Poźniak's theorem (Theorem 3.2) several times. Biran, Polterovich and Salamon [BPS], Niche [Ni] and Xue [Xu] also used Poźniak's theorem to get an upper bound of the Biran-Polterovich-Salamon capacity. In their papers, the homomorphism T [a,∞);c α is an isomorphism. However in our case, T [a,∞);c α is not an isomorphism. Thus we need more sophisticated arguments.
We define a capacity C introduced by the first author in [Ka] in terms of the Biran-Polterovich-Salamon capacity [BPS,Subsection 3.2] (see also [Ni, We, Xu]  On the other hand, the first author essentially showed Theorem 1.4 in [Ka]. Theorem 1.4 ( [Ka,Theorem 1.2]). For any R > 0 and ∈ Z we have C(T 2n , T n ; R, 0, , −∞) ≤ 2R| |.
Therefore Theorem 1.2 improves Theorem 1.4. Moreover, the first author proposed the following conjecture.
Theorem 1.2 proves Conjecture 1.5 for (M, X) = (T 2n , T n ). The paper is organized as follows. In Section 2, we introduce the Floer homology and the symplectic homology for non-contractible trajectories which are the main tools to prove our main theorems. In Section 3, we calculate the dimensions of the Floer homology and the symplectic homology to prove our main theorems. In Section 4, we prove our main theorems (Theorems 1.1 and 1.2).

Symplectic homology
In this section, we define the Floer homology for non-contractible periodic trajectories (see [BPS,Section 4] for details).
Let (M , ω) be a compact symplectic manifold with convex boundary ∂M (i.e., there exists a Liouville vector field defined on an open neighborhood of ∂M in M and pointing outward along ∂M ) and denote M = M \ ∂M . Although the product of compact symplectic manifolds with convex boundary need not have convex boundary, we can still define the Floer homology of the product according to [FS,Products,Section 3]. In Section 3, we will consider the Floer homology of the product A R × T 2n which has no convex boundary, where A R = [−R, R] × R/Z.
2.1. Action functional. For a free homotopy class α ∈ [S 1 , M ] denote by L α M the space of free loops S 1 → M representing α. In addition, we assume that our manifold (M , ω) is symplectically α-atoroidal, i.e., for any free loop u in L α M , that is for u : S 1 → L α M considered as the map u : T 2 → M from the two-torus, The Hamiltonian isotopy {ϕ t H } t∈[0,1] associated to H is defined by and its time-one map ϕ H = ϕ 1 H is referred to as the Hamiltonian diffeomorphism of H. Let P(H; α) be the set of one-periodic trajectories of ϕ H representing α. A one-periodic trajectory x ∈ P(H; α) is called non-degenerate if it satisfies det dϕ H (x(0)) − id = 0. Fix a reference loop z ∈ α. We define the action functional A H : where H t = H(t, ·) andx is a path in L α M between z and x considered as a map x : [0, 1] × S 1 → M from the annulus [0, 1] × S 1 to M . Since our manifold (M , ω) is symplectically α-atoroidal, the functional A H is well-defined as a real-valued function. Note that P(H; α) is equal to the set of critical points of A H .
We define the action spectrum of A H by Spec(H; α) = A H P(H; α) .
Let a and b be real numbers such that −∞ ≤ a < b ≤ ∞. Suppose that the Hamiltonian H satisfies a, b ∈ Spec(H; α) and that it is regular, i.e., every one- 2.2. Filtered Floer chain complex. We define the chain group of our Floer chain complex to be the Z/2Z-vector space Let J t = J t+1 ∈ J (M, ω) be a time-dependent smooth family of ω-compatible almost complex structures on M such that J t is convex and independent of t near the boundary ∂M . Consider the Floer differential equation Here we note that grad A H (u(s, ·)) = J t u(s, ·) ∂ t u(s, ·) − X Ht u(s, ·) for all s. For a smooth solution u to (1) we define the energy by the formula Then we have the following lemma.
(i) There exist periodic solutions x ± ∈ P(H; α) such that where both limits are uniform in the t-variable. (ii) The energy identity holds: We call a family of almost complex structures regular if the linearized operator for (1) is surjective for any finite-energy solution of (1) in the homotopy class α. We denote by J reg (H; α) the space of regular families of almost complex structures. This subspace is generic in J (M, ω) (see [FHS]). For any J ∈ J reg (H; α) and any pair x ± ∈ P(H; α) the space M(x − , x + ; H, J) = { solution of (1) satisfying (i) } is a smooth manifold whose dimension near such a solution u is given by the difference of the Conley-Zehnder indices (see [SZ]) of x − and x + relative to u.
The subspace of solutions of relative index 1 is denoted by M 1 (x − , x + ; H, J). For J ∈ J reg (H; α) the quotient M 1 (x − , x + ; H, J)/R is a finite set for any pair x ± ∈ P(H; α). We define the boundary operator ∂ H,J : where # 2 denotes the modulo 2 counting.

Continuation. We define the set
Proposition 2.5 ( [BPS,Remark 4.4.1]). Every Hamiltonian H ∈ H a,b (M ; α) has a neighborhood U such that the Floer homology groups HF [a,b) (H , J ; α), for any regular H ∈ U and any regular almost complex structure J ∈ J reg (H ; α), are naturally isomorphic.
According to Proposition 2.5, one can define the Floer homology HF [a,b) (H; α) whether H is regular or not.
Definition 2.6. For H ∈ H a,b (M ; α) we define HF [a,b) where H is any regular Hamiltonian sufficiently close to H.

Monotone homotopies. We introduce a bidirected partial order on
Then there exists a homotopy {H s } s from H 0 to H 1 such that ∂ s H s ≤ 0. We call such a homotopy of Hamiltonians monotone. Let α ∈ [S 1 , M ] be a nontrivial free homotopy class and a, b ∈ R ∪ {∞} such that a < b. It follows from the energy identity that the Floer chain map Φ H1H0 : CF(H 0 ; α) → CF(H 1 ; α), defined in terms of the solutions of the equation preserves the subcomplexes CF a (H 0 ; α) and CF b (H 0 ; α). Hence every monotone homotopy {H s } s induces a natural homomorphism [BPS,Subsection 4.5]). The homomorphism σ H1H0 is called the monotone homomorphism from H 0 to H 1 . We call a monotone homotopy {H s } s action-regular if H s takes values in a connected component of H a,b (M ; α).
H 1 , we obtain the following commutative diagram, whose rows are the short exact sequences for H 0 and for H 1 .
where ι F and π F denote the natural inclusion and projection, respectively. The associated long exact sequences induce the following commutative diagram. ( 2.5. Symplectic homology. In this subsection, we consider a homology introduced in [FH, CFH, Ci]. We refer to [BPS,Subsection 4.8] for details. Let α ∈ [S 1 , M ] be a nontrivial free homotopy class and a, b ∈ R ∪ {∞} such that a < b. As mentioned in Subsection 2.4, there is a natural homomorphism These homomorphisms define an inverse system of Floer homology groups over H a,b (M ; α), . We denote the symplectic homology of M in the homotopy class α for the action interval [a, b) by Fix a compact subset A ⊂ M and a constant c ∈ R. We define the set This defines a directed system of Floer homology groups over H a,b c (M, A; α), . We denote the relative symplectic homology of the pair (M, A) at the level c in the homotopy class α for the action interval [a, b) by Proposition 2.10 ( [BPS,Proposition 4.8.2]). Let α ∈ [S 1 , M ] be a nontrivial homotopy class and suppose that −∞ ≤ a < b ≤ ∞. Then for any c ∈ R there exists a unique homomorphism such that for any two Hamiltonians Here π H0 and ι H1 are the canonical homomorphisms. In particular, since σ HH = id for any H ∈ H a,b c (M, A; α), the following diagram commutes.
Remark 2.11. As we noted in Remark 2.7, we can still define the filtered symplectic homology for α = 0 M and the conclusion of Proposition 2.10 still holds if the action interval [a, b) does not contain zero.
The original version of Theorem 3.2 can be found in [Po,Theorem 3.4.11].
Remark 3.3. The grading of the Floer homology groups is well-defined up to an additive constant. More precisely, with a suitable choice of this grading, HF is isomorphic to HM dim C0−k (C 0 , f, g), and hence to H k (P ; Z/2Z) if P is orientable. In the present paper, we choose this grading for simplicity.

3.2.
Dimensions of symplectic homology groups. Let (N, ω N ) be a closed connected symplectic manifold and X ⊂ N a compact subset. Let A R denote the annulus (−R, R)×R/Z, and for u ∈ (−R, R) we define its subset We consider the product symplectic manifold A R × N, (dp 0 ∧ dq 0 ) ⊕ ω N and the free homotopy class (α , In this subsection, we give an explicit computation of symplectic homology groups in the case that (N, ω N ) = (T 2n , ω std ) = (R/2Z × R/Z) n , ω std and X = ({0} × R/Z) n ∼ = T n . In the following theorem, let 0 denote the trivial homotopy Theorem 3.4. Let R > 0 and u be real numbers such that u ∈ (−R, R). Then for any a > 0 and any c > 0, we have For = 0, we have the following result. For the sake of brevity, we put Theorem 3.5. Let R > 0 and u be real numbers such that u ∈ (−R, R), and ∈ Z \ {0}. Then for any a ∈ R and c > max{u , 0}, we have Theorem 3.6. Let R > 0 and u be real numbers such that u ∈ (−R, R) and let ∈ Z. The homomorphism is non-zero if and only if R| | < a ≤ c − u and k = 0, 1, . . . , n + 1. Moreover, in this case, the homomorphism T [a,∞);ĉ α k is surjective and We put m u = min{1, R − |u|} and u = (u 0 , u 1 , . . . , u n ) = (u, 0, . . . , 0) ∈ (−R, R) × (R/2Z) n .

Proof of Theorem 3.4.
Proof. The proof of [BPS,Theorem 5.1.1] carries over almost literally. Fix a positive real number c > 0 and choose a smooth family of real functions {f s (r)} s∈R , defined for r ∈ R, with the following properties (cf. [BPS,Subsection 5.4]).
(i) f s (−r) = f s (r) for all s and r.
(ii) For any s ∈ R (iii) For all s and r we have ∂ s f s (r) ≥ 0.
(iv) If s ≥ 1, then We note that every contractible trajectory in A R × T 2n is constant. By (vi), for s ≥ 1 the set P(H s ; 0) of contractible periodic trajectories is denoted by and for s ≤ −1 P(H s ; 0) = t → p(t), q(t) q 0 ≡q 1 ≡ · · · ≡q n ≡ 0, By (ii) and Lemma 3.7, the set P(H s ; 0) is a Morse-Bott manifold of periodic trajectories for H s . Moreover, P(H s ; 0) ∼ = T 2n+1 for s ≥ 1 and P(H s ; 0) ∼ = T n+1 s ≤ −1. For every s ∈ R, the action of x ∈ P(H s ; 0) is By Theorem 3.2, for every s ∈ R we have By [BPS,Proposition 4.5.1], the monotone homomorphism is an isomorphism whenever a ∈ [f s1 (0), f s0 (0)] and either 1 ≤ s 1 ≤ s 0 or s 1 ≤ s 0 ≤ −1. By [BPS,Lemma 4.7.1 (ii)], the homomorphism is an isomorphism for any s ≥ 1 such that f s (0) > a. Therefore, By [BPS,Lemma 4.7.1 (i)], the homomorphism is an isomorphism for any s ≤ −1 if a ≤ c, and for any s ≤ −1 with f s (0) < a if a > c. Hence we obtain Thus the proof of Theorem 3.4 is complete.
Proof. We assume, without loss of generality, that > 0. (iii) For all s and r we have ∂ s f s (r) ≥ 0.
(vi) For any s ≥ 1 such that f s (0) > R there exist real numbers r s < r s < 0 such that and f s (r) = R for any r ∈ R \ {r s , r s }.

Now choose a family of Hamiltonians
Here we note that the condition m u ≤ 1 ensures that the Hamiltonians H s , s ≤ −1 are well-defined. Then for s ≥ 1 the corresponding Hamiltonian vector field X Hs is of the form For ∈ Z >0 let us now consider the nontrivial free homotopy clasŝ Then for s ≥ 1 the set P(H s ;α ) of periodic trajectories inα is denoted by and for s ≤ −1 where c 0 ∈ R/Z, q 1 ≡ c 1 , q 2 ≡ c 2 , · · · , q n ≡ c n ∈ R/Z, p 0 (t) = u + m u r, where r < 0 is such that f s (r) = m u , p 1 = p 2 = · · · = p n = 0, and (5) Note that there are no periodic trajectories representingα if p 0 ≤ −(R + |u|)/2 and s ≤ −1. Given r < 0 with f s (r) = R , we denote Given r < 0 with f s (r) = m u , we denote Given r > 0 with f s (r) = (R − |u|) , we denote Then P(r,α ) and R(r,α ) are diffeomorphic to T 2n+1 , and Q(r,α ) is diffeomorphic to T n+1 . In summary, we have We claim that the value of the action functional on R(r,α ) is negative. In fact, since we have f s (r) = (R − |u|) for any periodic trajectory in R(r,α ), the action of such a periodic trajectory is f s (r) − |u| − (R − |u|)r , and this is negative by (viii). On the other hand, by (vi), (vii) and Lemma 3.8, P(r,α ) and Q(r,α ) are Morse-Bott manifolds of periodic trajectories for H s and the values of the action functional on these critical manifolds are A Hs P(r,α ) = f s (r) − Rr , and A Hs Q(r,α ) = f s (r) − (u + m u r) , respectively. Fix a real number a. We prove Theorem 3.5 in four steps.
Step 1. If a < R , then SH For s ≥ 1 the only families of periodic trajectories are P(r s ,α ) and P(r s ,α ). Since both r s and r s converge to −1 as s → ∞, the values of the action functional on P(r s ,α ) and P(r s ,α ) are both bigger than a for s sufficiently large. Hence for s sufficiently large. Here since HF [−∞,∞) (H s ;α ) is independent of H s (see [BPS,Proposition 4.5.1]) and every C 2 -small Hamiltonian has only contractible periodic trajectories, the last equation holds. Now Step 1 follows from [BPS,Lemma 4.7.1 (ii)].
Step 2. If a ≥ R , then SH is an isomorphism whenever f s (0) > a.
By (vi), for any s ≥ 1 the number r s (resp. r s ) is the maximum point (resp. the minimum point) of the function f s, : If s is sufficiently large so that f s (0) > a, then f s, (0) = f s (0) > a and hence is an isomorphism whenever f si (0) > a for i = 0, 1 and s 1 ≤ s 0 and Step 2 follows from [BPS,Lemma 4.7.1 (ii)].
Step 3. If a > c − u > 0, then SH In Figure 4, the lines L 1 , L 2 are tangent to H s at p 0 = |u| + (R − |u|)r, where r > 0 is such that f s (r) = (R − |u|) (if such r > 0 exists). Since a > 0, we can ignore all trajectories which belong to R(r,α ). Since both r s and r s converge to 0 as s → −∞, the values of the action functional on Q(r s ,α ) and Q(r s ,α ) are both less than a for −s sufficiently large. Hence HF [a,∞) (H s ;α ) = 0 for −s sufficiently large. Now Step 3 follows from [BPS,Lemma 4.7.1 (i)].
Moreover, the homomorphism is an isomorphism for s −1.
As a > 0 we may ignore trajectories which belong to R(r,α ). By (vii), for any s ≤ −1 the number r s (resp. r s ) is the maximum point (resp. the minimum point) of the function g s, : [−1, 0] → R defined by Then, by (ii), g s, (0) = f s (0) − u > c − u ≥ a and hence A Hs Q(r s ,α ) = g s, (r s ) > g s, (0) > a.
Proof. If the homomorphism is non-zero, then Theorem 3.4, Step 1 and Step 3 in the proof of Theorem 3.5 imply that R| | < a ≤ c − u . Fix R| | < a ≤ c − u . For simplicity, we assume that = 0. The proof for the case = 0 follows the same path as in the case = 0. We choose a sufficiently large T 1 so that for all k the homomorphisms and are isomorphisms. By Proposition 2.10, we have the following commutative diagram.
Hence it is enough to show that the homomorphism σ H −T H T k is non-zero and surjective.
Thus for any x ∈ P(r T ,α ), We choose a positive real number b such that Since the minimum value of F T is −n(n + 2), we obtain a < A H T P(r T ,α ) − εn(n + 2) < b. We deform H T (ερ T F T )| A R ×T 2n \U small enough so that every periodic trajectory x lying in A R ×T 2n \U has action less than a (i.e., the tangent line at x of slope does not take values greater than a at p 0 = 0), where U is a small open neighborhood of { p 1 = · · · = p n = 0 }. We then obtain a Hamiltonian H : A R × T 2n → R with compact support so that H = H T (ερ T F T ) near p 1 = · · · = p n = 0, and the Hamiltonian isotopy associated to H does not admit periodic trajectories of action greater than a and less than b. p 0 p 1 Figure 6. Outline of the graph of H in the case n = 1. assume, without loss of generality, that i 0 = n), then we have This contradicts the fact that F T (c) > −1. Hence S is homeomorphic to the product of a small open n-cell e n and the (n + 1)-torus T n+1 . Theorem 3.2 and Claim 1 show that the Floer homology HF is an isomorphism.
Proof. We can choose a monotone homotopy {H s } s defining the map σ H H T so that H s does not allow new periodic trajectories of action greater than b for all s, i.e., {H s } s is action-regular. Hence Lemma 2.9 shows that σ H H T is an isomorphism. is an isomorphism.
Proof. Since the Hamiltonian isotopy associated to H does not admit periodic trajectories of action greater than a and less than b, we have HF [a,b) ( H;α ) = 0. The exactness of the second row in (6) shows Claim 4. Actually, Step 4 directly shows that HF [a,∞) ( H −T ;α ) ∼ = H * (T n+1 ; Z/2Z). By Claim 3-5 and (7), we deduce that the homomorphism is non-zero and surjective if and only if so is the homomorphism Therefore, the exactness of the first row in (6) implies that it is suffices, for proving the non-zeroness of [π F ] k , to show that the homomorphism is surjective if R| | < a ≤ c − u and k = 0, 1, . . . , n + 1. Thus the proof of Theorem 3.6 is complete.

Annular capacity
To prove Theorem 1.1, we introduce a homological "annular" capacity. This capacity is defined in terms of the homological Biran-Polterovich-Salamon capacity [BPS].
Here we use the convention that inf ∅ = ∞ and sup ∅ = −∞. Let (N, ω N ) be a closed connected symplectic manifold and X ⊂ N a compact subset. Let A R denote the annulus (−R, R) × R/Z, and for u ∈ (−R, R) define its subset Definition 4.2. For R > 0, u ∈ (−R, R), ∈ Z and a ≥ −∞, we define a homological relative capacity C(N, X; R, u, , a) by C(N, X; R, u, , a) = C BPS (A R × N, L u × X; (α , 0 N ), a).
Here A R ×N is considered as the symplectic manifold equipped with the product symplectic form (dp 0 ∧ dq 0 ) ⊕ ω N where (p 0 , q 0 ) ∈ A R = (−R, R) × R/Z.

4.2.
Proof of Theorem 1.1. In the case that N = T 2n and X = T n , we can compute the capacity C directly due to Theorem 3.6. Proposition 4.3. For any R > 0, u ∈ (−R, R), ∈ Z and a ≥ −∞, C(T 2n , T n ; R, u, , a) = max{R| | + u , a + u }.
is non-zero if and only if R| | < s ≤ c − u . Hence we have A c (A R × T 2n , L u × T n ;α ) = (R| |, c − u ] for any c > 0 such that R| | ≤ c − u . Therefore, for any a ≥ −∞ we have C(T 2n , T n ; R, u, , a) = inf c > 0 sup A c (A R × T 2n , L u × T n ;α ) > a = max{R| | + u , a + u }.
Proposition 4.4 relates the capacities C(N, X; R, u, , a) and C(N, X; R, u, , a).
Proposition 4.4 ( [BPS,Proposition 4.9.1]). Let R > 0 and u be real numbers such that u ∈ (−R, R). Let be an integer and a a real number. If C(N, X; R, u, , a) < ∞ then every Hamiltonian H with compact support on S 1 ×A R ×N with H| S 1 ×Lu×X ≥ C(N, X; R, u, , a) has a one-periodic trajectory x in the homotopy class (α , 0 N ) with the action A H (x) ≥ a. In particular, C(N, X; R, u, , a) ≥ C(N, X; R, u, , a).
Proof of Theorem 1.2. By Proposition 4.3 and Proposition 4.4, it is enough to show that C(T 2n , T n ; R, u, , a) ≥ max{R| | + u , a + u }.
If = 0 and a ≤ 0, then every Hamiltonian H with compact support has a contractible periodic trajectory whose action is zero and hence C(T 2n , T n ; R, u, 0, a) = 0. Thus we assume that either = 0 and a > 0, or = 0. We set m = max{R| | + u , a + u }. For any δ > 0 choose a smooth function f : (−R, R) → R with compact support satisfying f (r) = m − δ for r near u, 0 for r near ± R, Now consider the Hamiltonian H : A R × T 2n → R with compact support given by H(p, q) = f (p 0 ). Then every periodic trajectory x : t → x(t) = p(t), q(t) of H in the homotopy class (α , 0 T 2n ) ∈ [S 1 , A R × T 2n ] satisfieṡ q 0 ≡ ,q 1 ≡ · · · ≡q n ≡ 0, p 0 (t) = r,ṗ 1 ≡ · · · ≡ṗ n ≡ 0, where r ∈ (−R, R) is such that f (r) = . Moreover, the action of x is A H (x) = f (r) − r . If R| | ≥ a, then we have m = R| | + u and hence f (r) Here we note that ≥ 0 (resp. < 0) if and only if r ≤ u (resp. r > u). The above observation contradicts the fact that f (r) = . Hence there is no one-periodic trajectory of length | |. We assume that R| | < a. Then m = a + u . If ≥ 0, then A H (x) = f (r) − r < R + r R + u m − r = (R + r)a + (u − r)R R + u < (R + r)a + (u − r)a R + u = a.
If < 0, then Thus there is no periodic trajectory in (α , 0 T 2n ) whose action is at least a. As a conclusion, we obtain that C(T 2n , T n ; R, u, , a) = C BPS (A R × T 2n , L u × T n ; (α , 0 T 2n ), a) ≥ m − δ for any δ > 0. It means that C(T 2n , T n ; R, u, , a) ≥ m.
Finally we prove Theorem 1.1.
Proof of Theorem 1.1. By [BPS, Proof of Theorem B], we may assume, without loss of generality, that H is one-periodic in time. If = 0 and a ≤ 0, then every Hamiltonian H with compact support has infinitely many contractible periodic trajectories whose actions are zero, and hence Theorem 1.1 is proved. Thus we assume that either = 0 and a > 0, or = 0. According to Theorem 1.2, we have C(T 2n , T n ; R, u, , c − u ) = max{R| | + u , c} = c.
It implies that for all H ∈ H c (A R × T 2n , L u × T n ) there exists x ∈ P(H; (α , 0 T 2n )) such that A H (x) ≥ c − u . Moreover, by Theorem 3.6, if H is non-degenerate, then the number of such x's is at least