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Given periodic orbits x1 ∈ P(H1), x2 ∈ P(H2), and y ∈ P(H1#H2), the space M Λ Υ (x1, x2; y)
consists of maps
u : Σ Λ Υ → T ∗ M,
solving the Floer equation ∂J,H(u) = 0, with asymptotics
lim
s→−∞ u(s, t − 1) = x1(t), lim
s→−∞ u(s, t) = x2(t), lim u(s, 2t − 1) = y(t).
s→+∞
We can associate to a map u : Σ Λ Υ → T ∗ M the map v : Σ :=Ê×[0, 1] → T ∗ M 2 by setting
v(s, t) := � −u(s, −t), u(s, t) � .
The identifications (153) on the left-hand side of the domain of u are translated into the fact that
v(s, t) is 1 periodic in t for s ≤ 0, or equivalently into the nonlocal boundary condition
(v(s, 0), −v(s, 1)) ∈ N ∗ ∆ M 2 ∀s ≤ 0, (155)
where ∆ M 2 denotes the diagonal in M 4 = M 2 × M 2 . The identifications (154) on the right-hand
side of the domain of u are translated into the local boundary conditions
v(s, 0) ∈ N ∗ ∆M, v(s, 1) ∈ N ∗ ∆M ∀s ≥ 0. (156)
The map u solves the Floer equation ∂J,H(u) = 0 if and only if v solves the Floer equation
∂J,K(v) = 0, where K :Ì×T ∗ M 2 →Êis the Hamiltonian
K(t, x1, x2) := H1(−t, −x1) + H2(t, x2) ∀(t, x1, x2) ∈Ì×T ∗ M 2 ,
which satisfies the growth conditions (H1) and (H2). The asymptotic conditions for u are equivalent
to
lim
s→−∞ v(s, t) = (−x1(−t), x2(t)), lim
s→+∞ v(s, t) = � −y((1 − t)/2), y((1 + t)/2) � . (157)
Finally, since
|∇v(s, t)| 2 = |∇u(s, −t)| 2 + |∇u(s, t)| 2 ,
the energy of u equals the energy of v,
�
E(u) := |∇u| 2 �
dsdt =
Σ Λ Υ
Σ
|∇v| 2 dsdt =: E(v).
We conclude that M Λ Υ (x1, x2; y) can be identified with the space of maps v : Σ → T ∗M2 which
solve the Floer equation for the Hamiltonian K, satisfy the boundary conditions (155), (156), and
the asymptotic conditons (157).
By Nash theorem, we can find an isometric embedding M ֒→ÊN for N large enough. The
induced embedding M2 ֒→Ê2N is such that
∆M2 = M 2 ∩ ∆Ê2N, ∆M × ∆M = M 2 ∩ � �
∆ÊN × ∆ÊN .
Note that the linear subspaces ∆Ê2N and ∆ÊN × ∆ÊN are partially orthogonal inÊ4N . Therefore,
the problem M Λ Υ reduces to the above setting, with Q = M2 , k = 1, s1 = 0, Q0 = ∆ M 2,
Q1 = ∆M × ∆M, V0 = ∆Ê2N, and V1 = ∆ÊN × ∆ÊN.
The energy of solutions in M Λ Υ (x1, x2; y) is bounded above from
�H1(x1) +�H2(x2) −�H1#H2(y),
so Proposition 6.2 implies that these solutions have a uniform L ∞ bound.
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