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with non-local boundary conditions
⎧
⎨ N
(u(s, 0), −u(s, 1)) ∈
⎩
∗∆M2 if s ≤ 0,
N ∗∆Θ M if 0 ≤ s ≤ α,
N ∗ (∆M × ∆M) if s ≥ α,
and asymptotics
(164)
lim
s→−∞ u(s, t) = � −x1(−t), x2(t) � , lim
s→+∞ u(s, t) = � −z((1 − t)/2), z((1 + t)/2) � . (165)
Similarly, we can view the space M Λ Υ (x1, x2; z) as the space of maps u : Σ → T ∗M 2 solving the
equation (163) with asymptotics (165) and non-local boundary conditions
(u(s, 0), −u(s, 1)) ∈
� N ∗ ∆M 2 if s ≤ 0,
N ∗ (∆M × ∆M) if s ≥ 0.
(166)
Compactness. We start with the following compactness results, which also clarifies the sense
of the convergence in (ii):
6.6. Lemma. Let (αh, uh) be a sequence in M Υ GE (x1, x2; z) with αh → 0. Then there exists
u0 ∈ M Λ Υ (x1, x2; z) such that up to a subsequence uh converges to u0 in C∞ loc (Σ \ {0, i}), in
C∞ (Σ ∩ {|Re z| > 1}), and uniformly on Σ.
Proof. Since the sequence of maps (uh) has uniformly bounded energy, Proposition 6.2 implies a
uniform L ∞ bound. Then, the usual non-bubbling-off analysis for interior points and boundary
points away from the jumps in the boundary condition implies that, modulo subsequence, we have
uh → u0 in C ∞ loc (Σ \ {0, i}, T ∗ M 2 ),
where u0 is a smooth solution of equation (163) on Σ \ {0, i} with bounded energy and satisfying
the boundary condions (166), except possibly at 0 and i. By Proposition 6.5, the singularities
0 and i are removable, and u0 satisfies the boundary condition also at 0 and i. By the index
formula (162) and transversality, the sequence uh cannot split, so u0 satisfies also the asymptotic
conditions (165), and uh → u0 in C ∞ (Σ ∩ {|Rez| > 1}). Therefore, u0 belongs to M Λ Υ (x1, x2; z),
and there remain to prove that uh → u uniformly on Σ.
We assume by contraposition that (uh) does not converge uniformly on Σ. By Ascoli-Arzelà
theorem, there must be some blow-up of the gradient. That is, modulo subsequence, we can find
zh ∈ Σ converging either to 0 or to i such that
Rh := |∇uh(zh)| = �∇uh�∞ → ∞.
For sake of simplicity, we only consider the case where zh = (sh, 0) → 0, 0 < sh < αh → 0. The
general case follows along analogous arguments using additional standard bubbling-off arguments.
For more details, see e.g. [HZ94, section 6.4].
We now have to make a case distinction concerning the behaviour of the quantity 0 < Rh ·αh <
∞:
(a) The case of a diverging subsequence Rhj · αhj → ∞ can be handled by conformal rescaling
vj(s, t) := uhj(shj +s/Rhj, t/Rhj) which provides us with a finite energy disk with boundary
on a single Lagrangian submanifold of conormal type. This has to be constant due to the
vanishing of the Liouville 1-form on conormals, contradicting the convergence of |∇vj(0)| = 1.
(b) The case of convergence of a subsequence Rhj · αhj → 0 can be dealt with by rescaling
vj(s, t) := uhj(shj + αhj, αhjt). Now vk has to converge uniformly on compact subsets
towards a constant map, since �∇vj�∞ = |∇vj(0)| = Rhj · αhj → 0. This in particular
implies that uhj(·, 0) |[0,αh j ] converges uniformly to a point contradicting the contraposition
assumption.
94