MATEMATIQKI VESNIK

59 (2007), 211–228

UDK 515.143

originalni nauqni rad research paper

GLUING AND PIUNIKHIN-SALAMON-SCHWARZ ISOMORPHISM FOR LAGRANGIAN FLOER HOMOLOGY

Jelena Katić

Abstract. We prove Floer gluing theorem in the case of objects of mixed type, that incorporate both Morse gradient trajectories and holomorphic discs with Lagrangian boundary conditions.

1. Introduction

Let M be a smooth compact manifold of dimension n and f: M → R a Morse function. Let P = T^{∗}M be a cotangent bundle over M, L_{0} = O_{M} a zero section, H: T^{∗}M → R compactly supported Hamiltonian and L_{1} = ϕ_{1}^{H}(L_{0}) a Hamiltonian deformation of O_{M}. Denote by CM_{∗}(f) Morse chain groups generated by the critical points of f and by CF_{∗}(H) Floer chain groups generated by the set L_{0}∩L_{1} (both with Z_{2}-coefficients). Let HM_{∗}(f) and HF_{∗}(H) be the corresponding Morse and Floer homology groups (the latter are well defined in this situation). The isomorphism between HM_{∗}(f) and HF_{∗}(H) was constructed in [9], following Piunikhin, Salamon and Schwarz’s construction for the case of periodical orbits [16]. The purpose of [9] was to prove that isomorphisms in Floer homology for Lagrangian intersections naturally intertwine with analogous isomorphisms in Morse homology. The isomorphism constructed there was based on counting the objects of mixed type. More precisely, we are interested in the following three moduli spaces. For p and q two critical points of Morse function f, denote by M(p, q, f) the set of negative gradient flows of f, i.e. solutions of equation:

\{ \begin{matrix}\frac{dγ}{ds} + ∇f(γ) = 0 \\ γ(−∞) = p, γ(+∞) = q. \end{matrix}  (1)

For two Hamiltonian paths x and y that begin and end at O_{M}, let M(x, y, H) be the set of solutions of:

AMS Subject Classification: Primary 53D40, Secondary 57R58, 53D12.

Keywords and phrases: Lagrangian submanifolds; Floer homology; Morse theory; gluing.

This work is partially supported by Ministry of Science and Environmental Protection of Republic of Serbia Project #144020.

\{ \begin{matrix}\frac{∂u}{∂s} + J(\frac{∂u}{∂t} − X_{H}(u)) = 0 \\ u(s, i) ∈ L_{0}, i ∈ \{0, 1\} \\ u(−∞, t) = x(t), \\ u(+∞, t) = y(t). \end{matrix}  (2)

The equation (2) is the negative gradient flow of the action functional

A_{H}: Ω_{0} → R, A_{H}(α) := ∫_{0}^{1} α^{∗}θ − H_{t}(α(t)) dt

(where θ is the Liouville 1− form on T^{∗}M) defined on

Ω_{0} : = {α: [0, 1] → T^{∗}M | α(0), α(1) ∈ O_{M}}.

Denote by M̂(p, q, f) and M̂(x, y, H) these sets modulo R− actions γ(·) ↦ γ(·+τ), and u(·, ·) ↦ u(· + τ, ·).

Now consider the space of pairs (γ, u)

γ: (−∞, 0] → M, u: [0, +∞) × [0, 1] → T^{∗}M

that satisfy

\{ \begin{matrix}\frac{dγ}{ds} = −∇f(γ(s)), \\ \frac{∂u}{∂s} + J(\frac{∂u}{∂t} − X_{ρ_{R}H}(u)) = 0 \\ u(0, t), u(s, 0), u(s, 1) ∈ O_{M}, \\ γ(−∞) = p, u(+∞, t) = x(t), \\ γ(0) = u(0, \frac{1}{2}). \end{matrix}  (3)

Here ρ_{R}: [0, +∞) → R is a smooth function such that, for a fixed R,

ρ_{R}(s) = \{ \begin{matrix}1, & s ≥ R + 1 \\ 0, & s ≤ R. \end{matrix}

Denote by M(p, f; x, H) the set of solutions of (3) (see Fig. 1).

Fig. 1. Mixed object from M(p, f; x, H)

Gluing and Piunikhin-Salamon-Schwarz isomorphism for Lagrangian Floer homology 213

If m_{f}(p) is the Morse index of critical point p and μ_{H}(x) the Maslov index of Hamiltonian path x (see[2, 17, 18]) for definition of Maslov index, [14] for its application to grading of Floer homology groups and [12] for generalizations), then dim M(p, q, f) = m_{f}(p) − m_{f}(q) and dim M(x, y, H) = μ_{H}(x) − μ_{H}(y). In [9] we derived the following Proposition by relying upon the convergence results given in [8] and gluing results that are the subject of this paper.

Proposition 1. [9] For generic f and H, M(p, f; x, H) is a smooth manifold of dimension m_{f}(p) − (μ_{H}(x) + \frac{n}{2}). When m_{f}(p) = (μ_{H}(x) + \frac{n}{2}), it is compact, hence a finite set. When m_{f}(p) = (μ_{H}(x) + \frac{n}{2}) + 1 then we have the following identification for the boundary of one-dimensional manifold M(p, f; x, H):

\begin{aligned}∂M(p, f; x, H) & = ⋃_{m_{f} (q)=m_{f} (p)−1} M̂(p, q, f) × M(q, f; x, H) ∪ \\ & ∪ ⋃_{μ_{H}(y)=μ_{H}(x)+1} M(p, f; y, H) × M̂(y, x, H). \end{aligned}  (4)

A sketch of the proof of Proposition 1 is the following: let W^{u}(p, f) be the unstable manifold of the critical point p of a Morse function f and let W^{s}(x, H) be the set of solutions of

\{ \begin{matrix}u: [0, +∞) × [0, 1] → T^{∗}M \\ \frac{∂u}{∂s} + J(\frac{∂u}{∂t} − X_{ρ_{R}H}(u)) = 0 \\ u(0, t), u(s, 0), u(s, 1) ∈ O_{M}, \\ u(+∞, t) = x(t), \end{matrix}  (5)

Then dim W^{u}(p, f) = m_{f}(p) (see [13]) and dim W^{s}(x, H) = −μ_{H}(x) + \frac{n}{2} (see Appendix in [15]). For generic f, H the evaluation map

ev: W^{u}(p, f) × W^{s}(x, H) → M × M, (γ, u) ↦ (γ(0), u(0, \frac{1}{2}))

is transversal to the diagonal Δ and therefore M(p, f; x, H) = ev^{−1}(Δ) is a smooth manifold of codimension n in W^{u}(p, f) × W^{s}(x, H), so we obtain the dimension formula.

Let CM_{k}(f) be the set of all critical points of Morse index k and CF_{k}(H) the set of all Hamiltonian paths with ends in zero section of Maslov index μ_{H}(x) + \frac{n}{2}. For m_{f}(p) = μ_{H}(x) + \frac{n}{2} we denote the cardinal number (mod 2) of M(p, f; x, H) by n(p, f; x, H) and define homomorphism

Φ: CM_{k}(f) → CF_{k}(H) by p ↦ ∑_{μ_{H}(x)=m_{f} (p)− \frac{n}{2}} n(p, f; x, H)x  (6)

on the generators. It follows from Proposition 1 that the homomorphism Φ is well defined and that it is well defined also on HM_{∗}(f) (see [16, 9]). By using the analysis of the boundary of some auxiliary one-dimensional mixed moduli spaces (see [9]), one can prove that Φ is an isomorphism and that the diagram

HF_{∗}(H^{α}) ⟶^{S^{αβ}} HF_{∗}(H^{β})
Φ^{α}↑  ↑_{Φ^{β}}
HM_{∗}(f^{α}) ⟶_{T^{αβ}} HM_{∗}(f^{β})

commutes (see [9]). Here S^{αβ} and T^{αβ} are the natural isomorphisms in, respectively, Floer and Morse theory.

Piunikhin-Salamon-Schwarz morphisms in more general cases were investigated by Albers in [1] where it was shown that it does not have to be an isomorphism in general, Barraud and Cornea in [3], Leclercq in [11], Lalonde in [10] and others.

The crucial analytical tools used in the proof of Proposition 1 (and the similar characterizations of the boundary of moduli spaces) are Gromov compactness and gluing techniques. From Gromov compactness it follows that the sequence of mixed objects from M(p, f; x, H) (if it does not converge to the object from M(p, f; x, H)) must converge to a broken object. It proves one inclusion in (4). Gluing gives the opposite: it assigns to a broken object the sequence of non-broken mixed objects that converges (in the Gromov weak sense) toward it.

In this paper we carry out the proof of gluing theorem for the case of the mixed type objects. The natural idea that occurs is to reduce the gluing construction to the Morse (or Floer) case, such that the holomorphic strip (or gradient trajectory) part in mixed object stays fixed. Indeed, this is the idea that we use in the construction of the pre-glued objects. But then we confront the problem of the attaching point: the exact solutions of combined (ordinary and partial differential) equation that converge to the previously given, broken one, do not have the same fixed attaching point, but only the ends (Morse critical point at one end and Hamiltonian path, at the other). Thus we choose the point of view that encircles Morse and Floer trajectories, so first establish the analytical setting and construct Banach manifolds of mixed-type mappings. The tangent spaces to these manifolds will be Sobolev W^{1,r} and L^{r} spaces, for r > 2. Since we use some Hilbert space techniques, it would be more convenient to take W^{1,2} and L^{2}, as it can be done in Morse case (see [19]). But we deal with two-dimensional domains (such as for u in (3), so these maps do not have to be continuous, which is the case for r > 2, due to the Sobolev Embedding Theorem.

We would like to thank D. Milinković for several helpful discussions during the preparation of this paper.

2. Notation and results

We state the main result in the following

Theorem 2. a) Let K be a compact subset of M(p, f; x, H) × M̂(x, y, H). Then there exists a lower parameter bound ρ_{0} = ρ_{0}(K) and a smooth embedding

♯: K × [ρ_{0}, +∞) ↪ M(p, f; y, H)
((γ, u), v, ρ) ↦ (γ, u)♯_{ρ}v.

We call the map ♯ gluing of a mixed-type object (γ, u) and holomorphic strip v. For an arbitrary sequence of gluing parameters ρ_{n} → ∞, we have the weak convergence

(γ, u)♯_{ρ_{n}}v ⟶^{C_{loc}^{∞}} ((γ, u), v).

b) Let K be a compact subset of M̂(p, q, f) × M(q, f; x, H). Then there exists a lower parameter bound ρ_{0} = ρ_{0}(K) and a smooth embedding

♯: K × [ρ_{0}, +∞) ↪ M(p, f; x, H)
(α, (γ, u), ρ) ↦ α♯_{ρ}(γ, u).

The map ♯ is a gluing of a gradient trajectory α and mixed-type object (γ, u). For an arbitrary sequence of gluing parameters ρ_{n} → ∞, we have the weak convergence

α♯_{ρ_{n}}(γ, u) ⟶^{C_{loc}^{∞}} (α, (γ, u)).

Here we say that the sequence (γ_{n}, u_{n}) ∈ M(p, f; y, H) weakly converges to

the broken trajectory ((γ, u), v) ∈ M(p, f; x, H) × M̂(x, y; H) if (γ_{n}, u_{n}) ⟶^{C_{loc}^{∞}} (γ, u) and there is a sequence τ_{n} ∈ R, such that

u_{n}(· + τ_{n}, ·) ⟶^{C_{loc}^{∞}} v(·, ·).

Similarly, we say that the sequence (γ_{n}, u_{n}) ∈ M(p, f; y, H) weakly converges to the broken trajectory (α, (γ, u)) ∈ M̂(p, q, f)×M(q, f; x, H) if there is a sequence τ_{n} ∈ R, such that

γ_{n}(· + τ_{n}) ⟶^{C_{loc}^{∞}} α(·)

and (γ_{n}, u_{n}) ⟶^{C_{loc}^{∞}} (γ, u).

In order to abbreviate notations, we will assume w = (γ, u) through the rest of the paper.

Fig. 2. Gluing of broken trajectories

We will prove Theorem 2 in the next section. Before doing that, we introduce concepts and notation that we will use.

In order to analyze gluing of mixed objects, we need to equip the moduli spaces M(x, y, H) and M(p, f; x, H) with the smooth Banach manifold structure.

The topology and Banach manifold structure of M(x, y, H) is induced by the Sobolev norm in the following way. Let

D := R × [0, 1] ⊂ R^{2}.  (7)

For given Hamiltonian paths x(t) and y(t) with ends in O_{M}, denote by C^{∞}(x, y) the set of all smooth maps u that satisfy

\begin{matrix}u: D → T^{∗}M \\ u(s, 0), u(s, 1) ∈ O_{M} \\ u(−∞, t) = x(t), u(+∞, t) = y(t). \end{matrix}  (8)

The tangent space space T_{u}C^{∞}(x, y) to C^{∞}(x, y) at point u consists of all vector fields ξ such that

ξ: D → u^{∗}(T T^{∗}M),
ξ(s, t) ∈ T_{u(s,t)}T^{∗}M, ξ(s, 0), ξ(s, 1) ∈ T M,
ξ(−∞, t) = ξ(+∞, t) = 0.

For r > 2, let ∥ξ∥_{L^{r}} and ∥ξ∥_{W^{1,r}} stand for standard Sobolev norms

∥ξ∥_{L^{r}} = (∫∫_{D} |ξ|^{r} ds dt)^{\frac{1}{r}} , ∥ξ∥_{W^{1,r}} = (∫∫_{D} (|ξ|^{r} + |∇_{s}ξ|^{r} + |∇_{t}ξ|^{r}) ds dt)^{\frac{1}{r}} .

(9)

Denote by W_{u}^{1,r}(x, y) and L_{u}^{r}(x, y) the completions of this tangent space in W^{1,r}- and L^{r}-Sobolev norms. Finally, let P^{1,r}(x, y) be the space of all u that satisfy (8) such that the tangent space at u is given by

T_{u}P^{1,r}(x, y) = W_{u}^{1,r}(x, y).

The topology and Banach manifold structure of P^{1,r}(x, y) (hence the topology of M(x, y, H) ⊂ P^{1,r}(x, y)) is induced by the topology of W_{u}^{1,r}(x, y) by means of the exponential map

(exp_{u} ξ)(s, t) = exp_{u(s,t)} ξ(s, t)

(see [5] for details).

The set M(x, y, H) is a zero set of a smooth section:

K(u) = \overline{∂}_{H,J}(u) = \frac{∂u}{∂s} + J(u)\frac{∂u}{∂t} − J(u)X_{H}(u)  (10)

of a Banach bundle E(x, y) → P^{1,r}(x, y) with a fibre L_{u}^{r}(x, y) over u ∈ P^{1,r}(x, y) (see [4, 5]).

Its linearization at u is a Fredholm operator defined on W_{u}^{1,r}(x, y) with values in L_{u}^{r}(x, y). It has the form

DK_{u}(ξ) = ∇_{s}ξ + J(u)∇_{t}ξ + ∇_{ξ}J(u)∂_{t}u − ∇_{ξ}∇H(u)  (11)

and its index is equal to the difference of Maslov indices at the ends μ_{H}(x)−μ_{H}(y). In the local coordinates (10) is of the form

ζ(s, t) ↦ \frac{∂ζ}{∂s} (s, t) + J(s, t)\frac{∂ζ}{∂t} (s, t) + A(s, t)ζ(s, t).

The situation with M(p, f; x, H) is somewhat different. Although we initially defined it as a transversal intersection set of two manifolds, we need to consider it as the zero set of some Fredholm operator, in order to get one consistent picture including all three moduli spaces (gradient trajectories, holomorphic discs and mixed objects). Let p and x(t) be as before. For r > 2, denote by P^{1,r}(p) the Sobolev W^{1,r}-completion of the space of all smooth trajectories γ that satisfy:

\{ \begin{matrix}γ: (−∞, 0] → O_{M} \\ γ(−∞) = p, \end{matrix}

and by P^{1,r}(x) the Sobolev W^{1,r}-completion of the space of all smooth maps u that satisfy:

\{ \begin{matrix}u: [0, +∞) × [0, 1] → T^{∗}M \\ u(0, t), u(s, 0), u(s, 1) ∈ O_{M}, \\ u(+∞, t) = x(t). \end{matrix}

By Sobolev W^{1,r}-completion we assume the previous discussion: the completion of the tangent space and the topology induced by exponential map (for more details about this construction is Morse theory see [19]. We denote tangent spaces by

W_{γ}^{1,r}(p) = T_{γ}P^{1,r}(p), W_{u}^{1,r}(x) = T_{u}P^{1,r}(x).

Let ẽv be the following evaluation map:

ẽv: P^{1,r}(p) × P^{1,r}(x) → M × M, ẽv(γ, u) = (γ(0), u (0, \frac{1}{2})) .

For generic choices of a Morse function (or a Riemannian metric) ẽv is transversal to the diagonal Δ ⊂ M × M and

P^{1,r}(p, x) := ẽv^{−1}(Δ)

is infinite-dimensional smooth Banach submanifold of P^{1,r}(p)×P^{1,r}(x). As before, denote by W_{(γ,u)}^{1,r}(p, x) = T_{(γ,u)}P^{1,r}(p, x) the corresponding tangent space. The space M(p, f; x, H) is the zero set of a restriction of a smooth section

\begin{matrix}F = F̃|_{P^{1,r}(p,x)}, F̃ = (F_{1}, F_{2}), \\ F_{1}(γ) = \frac{dγ}{ds} + ∇f(γ), F_{2}(u) = \overline{∂}_{ρ_{R}H,J} u \end{matrix}  (12)

of a Banach bundle

E^{0,r}(p, x) → P^{1,r}(p) × P^{1,r}(x)  (13)

with a fibre L_{γ}^{r}(p) × L_{u}^{r}(x) over a point (γ, u) ∈ P^{1,r}(p) × P^{1,r}(x). Here as before, by L_{γ}^{r}(p) we denote the space of all ξ that satisfy

\{ \begin{matrix}ξ: (−∞, 0] → γ^{∗}(T M), \\ ξ(s) ∈ T_{γ(s)}M, ξ(−∞) = 0, \\ ∥ξ∥_{L^{r}} < ∞ \end{matrix}

and by L_{u}^{r}(x) the space of all ξ such that

\{ \begin{matrix}ξ: [0, +∞) × [0, 1] → u^{∗}(T T^{∗}M), ξ(s, t) ∈ T_{u(s,t)}T^{∗}M, \\ ξ(0, t), ξ(s, 0), ξ(s, 1) ∈ T M, ξ(+∞, t) = 0, \\ ∥ξ∥_{L^{r}} < ∞. \end{matrix}

The linearization of (12) at the point (γ, u) is

\begin{matrix}(DF)_{(γ,u)} : W_{(γ,u)}^{1,r}(p, x) → L_{γ}^{r}(p) × L_{u}^{r}(x), (DF)_{(γ,u)} = ((DF_{1})_{γ}, (DF_{2})_{u}), \\ (DF_{1})_{γ}: W^{1,r}(p) → L^{r}(p), (DF_{1})_{γ}η = ∇_{\frac{dγ}{ds}} η + ∇_{η}∇f(γ), \\ (DF_{2})_{u}: W^{1,r}(x) → L^{r}(x), \\ (DF_{2})_{u}ξ = ∇_{s}ξ + J(u)∇_{t}ξ + ∇_{ξ}J(u)∂_{t}u − ∇_{ξ}∇(ρ_{R}H)(u). \end{matrix} (14)

Proposition 3. The operator (14) is Fredholm hence the map (12) is a Fredholm map.

Proof. We will omit the subscripts u, γ, (γ, u) in order to abbreviate notations. First we observe that the manifold P^{1,r}(p, x) is of finite codimension in P^{1,r}(p) × P^{1,r}(x). Indeed, the tangent space W^{1,r}(p, x) of P^{1,r}(p, x) is the kernel of the differential Dẽv of the evaluation map. It holds:

W^{1,r}(p) × W^{1,r}(x)/ Ker(Dẽv) ≅ Im(Dẽv).

The image Im(Dẽv) is of finite dimension since the target space of Dẽv is so. It follows that W^{1,r}(p, x) is of finite codimension and from Hahn-Banach theorem that it can be complemented in W^{1,r}(p)×W^{1,r}(x) by some finite-dimensional space, denote it by X. Since codim_{M×M}(Δ) = n it holds dim X = n.

We consider the auxiliary operator

DF̃: W^{1,r}(p) × W^{1,r}(x) → L^{r}(p) × L^{r}(x)
DF̃ := (DF_{1}, DF_{2})

defined on the product W^{1,r}(p) × W^{1,r}(x) in order to compute the index of the operator DF in the terms of the indices of DF_{1} and DF_{2}. Since the operators DF_{1} and DF_{2} are Fredholm (see [19, 15]) and it holds

Ker DF̃ = Ker DF_{1} × Ker DF_{2}, Coker DF̃ = Coker DF_{1} × Coker DF_{2}

we conclude that DF̃ is also Fredholm with the Fredholm index Ind(DF̃) = Ind(DF_{1}) + Ind(DF_{2}). The operator DF is a restriction of DF̃ to the space W^{1,r}(p, x). Consider the following (disjoint) decompositions of the spaces W^{1,r}(p)× W^{1,r}(x) and L^{r}(p) × L^{r}(x):

W^{1,r}(p) × W^{1,r}(x) = X_{1} ⊕ X_{2} ⊕ X_{3} ⊕ X_{4}, L^{r}(p) × L^{r}(x) = Y_{1} ⊕ Y_{2} ⊕ Y

where the subspaces X_{i}, Y_{i} and Y are defined in the following way:

X_{3} := W^{1,r}(p, x) ∩ Ker(DF̃), X_{1} := W^{1,r}(p, x) ⊖ X_{3}
X_{4} := X ∩ Ker(DF̃), X_{2} := X ⊖ X_{4}
Y_{i} := DF̃(X_{i}), for i = 1, 2, Y := L^{r}(p) × L^{r}(x) ⊖ (Y_{1} ⊕ Y_{2}) .

(15)

The sign A ⊖ B stands for the complement of B in A and all the spaces in (15) are well defined due to the Hahn-Banach theorem. All the spaces except X_{1} and Y_{1} are of finite dimension: X_{3} and X_{4} are subspaces of Ker DF̃—which is of finite dimension; X_{2} is the subspace of X—which is of finite dimension; Y_{2} is the isomorphic image of X_{2}; finally Y is of finite dimension since it is a co-kernel of Fredholm operator DF̃. Set

m_{2} := dim(X_{2}) = dim(Y_{2}), m_{3} := dim(X_{3}), m_{4} := dim(X_{4}), m := dim(Y).

We see that

Ker(DF̃) = X_{3} ⊕ X_{4}, Coker(DF̃) = Y

and, since DF = DF̃|_{X_{1}⊕X_{3}} : X_{1} ⊕ X_{3} → Y_{1} ⊕ Y_{2} ⊕ Y,

Ker(DF) = X_{3}, Coker(DF) = Y_{2} ⊕ Y.

We conclude that DF is also Fredholm, and moreover, we compute its index:

Ind(DF) = dim(Ker(DF)) − dim(Coker(DF)) = dim(X_{3}) − dim(Y_{2} ⊕ Y)

= m_{3} − (m_{2} + m) = (m_{3} + m_{4}) − m − (m_{2} + m_{4})

= dim(Ker(DF̃)) − dim(Coker(DF̃)) − dim(X_{2} ⊕ X_{4})

= Ind(DF̃) − dim(X) = Ind(DF_{1}) + Ind(DF_{2}) − n

= m_{f}(p) − μ_{H}(x) − \frac{n}{2}.

3. Proof of Theorem 2

3.1. Pre-gluing. We first define a pre-gluing map, i.e. an approximate solution. Let w = (γ, u) ∈ M(p, f; x, H), i.e. γ ∈ W^{u}(p, f), u ∈ W^{s}(x, H) and v ∈ M(x, y, H). Denote by

β^{+} : R → [0, 1]  (16)

a smooth non-decreasing cut-off function, equal to 0 for s ≤ 0 and to 1 for s ≥ 1. Let u(s, t) = exp_{x(t)}(ξ(s, t)) for all t and s ≥ s_{0}, and v(s, t) = exp_{x(t)}(ζ(s, t)) for all t and s ≤ −s_{0}. For ρ ≥ max{2s_{0}, s_{0} + 1}, define

u ♯ _{ρ}^{0}v(s, t) := \{ \begin{matrix}u(s, t), & 0 ≤ s ≤ \frac{ρ}{2}, \\ exp_{x(t)}(β^{+}(−s + \frac{ρ}{2} + 1)ξ(s, t)), & \frac{ρ}{2} ≤ s ≤ \frac{ρ}{2} + 1, \\ x(t), & \frac{ρ}{2} + 1 ≤ s ≤ ρ \\ exp_{x(t)}(β^{+}(s − ρ)ζ(s − 2ρ, t)), & ρ ≤ s ≤ ρ + 1, \\ v(s − 2ρ, t), & s ≥ ρ + 1. \end{matrix}  (17)

Now set

ϖ := w ♯ _{ρ}^{0}v := (γ, u ♯ _{ρ}^{0}v).  (18)

The linearization of the pre-gluing map (17), for ξ ∈ Ker(DF_{2})_{u} = T_{u}W^{s}(x, H), ζ ∈ Ker(DK)_{v} = T_{v}M(x, y, H), is given by

ξˆ♯ζ := D♯_{ρ}^{0}(u, v)(ξ, ζ) = \{ \begin{matrix}ξ(s, t), & 0 ≤ s ≤ \frac{ρ}{2}, \\ ∇_{2} exp(β^{+}(−s + \frac{ρ}{2} + 1)∇_{2} exp^{−1} ξ(s, t)), & \frac{ρ}{2} ≤ s ≤ \frac{ρ}{2} + 1, \\ 0, & \frac{ρ}{2} + 1 ≤ s ≤ ρ \\ ∇_{2} exp(β^{+}(s − ρ)∇_{2} exp^{−1} ζ(s − 2ρ, t)), & ρ ≤ s ≤ ρ + 1, \\ ζ(s − 2ρ, t), & s ≥ ρ + 1. \end{matrix}

(19)

Here ∇_{2} exp is a fibre linearization of exponential map. More precisely, for P = T^{∗}M, let

K: T (T P) → T P, π: T P → P

denote the unique Levi-Civita connection with respect to the given Riemannian metric g and the canonical projection. For ξ ∈ T P in the injectivity neighborhood associated to exp, denote by:

∇_{1} exp(ξ) : = D exp(ξ) ∘ (Dπ|_{Ker(K(ξ))})^{−1} : T_{π(ξ)}P ⟶^{≅} T_{exp(ξ)}P
∇_{2} exp(ξ) : = D exp(ξ) ∘ (K|_{Ker(Dπ(ξ))})^{−1} : T_{π(ξ)}P ⟶^{≅} T_{exp(ξ)}P

(see Appendix A.2 in [19] for details).

The linearization of (18) is, when ς = (η, ξ) ∈ T_{w}M(p, f; x, H), i.e. η ∈ Ker D_{γ} = T_{γ}W^{u}(p, f),

ςˆ♯ζ = (η, ξ)ˆ♯ζ = (η, ξˆ♯ζ).

We will use the abbreviation χ for the triple (w, u, ρ) and (in the case when we want to emphasize the relation between ϖ and χ) the notation ϖ_{χ} for ϖ as in (18).

The local representation of the operator F from (12) with respect to ϖ gives rise to the bundle mapping

F_{ϖ}: ∇_{2} exp_{ϖ}^{−1} ∘F ∘ exp_{ϖ}: E^{1,r} ⊃ O_{ϖ} → E^{0,r}  (20)

where E^{1,r} and E^{0,r} are vector bundles over K × [ρ_{0}, ∞) with a fibres W_{ϖ}^{1,r}(p, x) and L_{γ}^{r}(p) × L_{u ♯_{ρ}^{0} v}^{r}(x) respectively. The fibre derivative

DF_{ϖ_{χ}}(0): W_{ϖ}^{1,r}(p, x) → L_{γ}^{r}(p) × L_{u ♯_{ρ}^{0} v}^{r}(x)

is exactly the linearization (DF)_{ϖ} at the point ϖ of a map F from (12).

We will use the abbreviation D_{χ} for DF_{ϖ_{χ}}(0). Similarly, we use the notation K_{u} for the local representation of the operator K from (10) and D_{u} for DK_{u}(0) = (DK)_{u}.

The exact solution of combined equation of type (3) will exist due to the Banach contraction principle (more precisely, its existence is the content of the abstract Lemma 5. The proof of the Theorem 2 is long and we will divide it in several steps. The first step is to prove that the linearized operator D_{χ} is onto for ρ large enough. This will imply the existence of the right inverse needed for the application of the abstract Lemma 5. It is done in the auxiliary Proposition 4. The second step is to prove that the mentioned right inverse is bounded and to check the other conditions required in Lemma 5—this is done in the Lemma 6 and in the rest of the Subsection 3.2. The last step is the proof of the embedding property which is given in the Subsection 3.3.

We will use the following notations. If ⟨·, ·⟩_{L^{2}} stands for standard scalar product:

⟨ξ, ζ⟩_{L^{2}} := ∫ ⟨ξ(τ), ζ(τ)⟩_{g} dτ

where g is Riemannian metric appearing in (9), then the following sets are well defined:

L̃_{w}^{⊥} := {η ∈ W_{w}^{1,r}(p, x) | ⟨η, ξ⟩_{L^{2}} = 0 for all ξ ∈ Ker D_{w}}
L̃_{v}^{⊥} := {η ∈ W_{v}^{1,r}(x, y) | ⟨η, ζ⟩_{L^{2}} = 0 for all ζ ∈ Ker D_{v}}
L_{χ}^{⊥} := {η ∈ W_{ϖ}^{1,r}(p, y) | ⟨η, ξ ˆ♯ζ⟩_{L^{2}} = 0 for all (ξ, ζ) ∈ Ker D_{w} × Ker D_{v}}.

Indeed, from the standard fact that the solutions of Morse gradient and Floer perturbed Cauchy-Riemann equations of types (1) and (2) have the exponential decay it follows that ξ ∈ L^{q}(p, x), ζ ∈ L^{q}(x, y) and ξ ♯ ζ ∈ L^{q}(p, y), where \frac{1}{r} + \frac{1}{q} = 1.

First we prove the following auxiliary Proposition.

Proposition 4. There is a lower parameter bound ρ_{1} such that for all gluing parameters ρ ≥ ρ_{1} and (w, v) ∈ K the Fredholm operator D_{χ}: W_{ϖ}^{1,r}(p, y) → L_{γ}^{r}(p)× L_{u ♯_{ρ}^{0} v}^{r}(y) of the type (14) is onto. Since Ker D_{χ} is of finite dimension, there is a complement Z of Ker D_{χ} in W_{ϖ}^{1,r}(p, y) and the projection onto Ker D_{χ}, denote it by Proj_{Ker D_{χ}}. Then the following composition induces an isomorphism:

φ_{χ} := Proj_{Ker D_{χ}} ∘ ˆ♯: Ker D_{w} × Ker D_{v} →^{≅} Ker D_{χ}.

Proof. From indices formulae

Ind(D_{w}) = m_{f}(p) − (μ_{H}(x) + \frac{n}{2})

Ind(D_{v}) = μ_{H}(x) − μ_{H}(y)

Ind(D_{χ}) = m_{f}(p) − (μ_{H}(x) + \frac{n}{2})

and the fact that D_{w} and D_{v} are onto, so

Ind D_{w} = dim Ker D_{w}, Ind D_{v} = dim Ker D_{v}

it follows

dim Ker D_{χ} ≥ Ind(D_{χ}) = Ind D_{w} + Ind D_{v} = dim Ker D_{w} + dim Ker D_{v}. (21)

It is sufficient to prove that, for fixed w, v ∈ K there exists a lower parameter bound ρ(w, v), such that, for ρ ≥ ρ(w, v) the map φ_{χ} is onto (since K is compact, this would imply the existence of the uniform lower bound ρ_{K} such that φ_{χ} is onto for any (w, v) ∈ K, ρ ≥ ρ_{K}). Indeed, if we suppose φ_{χ} is onto, then

dim Ker D_{w} + dim Ker D_{v} ≥ dim Ker D_{χ}.  (22)

From (21) and (22) we have

Ind D_{χ} = Ind D_{w} + Ind D_{v} = dim Ker D_{w} + dim Ker D_{v} = dim Ker D_{χ}

so the proof of the Proposition follows.

So we need to prove that φ_{χ} is onto. It is enough to show that for some C > 0 and ρ large enough it holds

∥D_{χ}ς∥_{L^{r}} ≥ C∥ς∥_{W^{1,r}}  (23)

for all ς ∈ L_{χ}^{⊥}. Indeed, if φ_{χ} is not surjective, it follows from the decomposition W_{ϖ}^{1,r}(p, y) = L_{χ}^{⊥} ⊕ Ker D_{w}ˆ♯Ker D_{u} that there exists ς ∈ Ker D_{χ} such that ς ∈ L_{χ}^{⊥}. But from (23) it follows that ς must be the zero vector. Suppose (23) is not true. Then there exist sequences ρ_{n} → ∞, ς_{n} ∈ L_{χ_{n}}^{⊥} such that

∥ς_{n}∥_{W^{1,r}} = 1 and ∥D_{χ_{n}}ς_{n}∥_{L^{r}} → 0.  (24)

Here ς_{n} = (w, v, ρ_{n}) χ_{n} = (w, v, ρ_{n}) and ϖ_{n} = ϖ_{χ_{n}}, i.e w and v are fixed, and ρ_{n} → ∞.

We can assume we are working with trivial case R^{2n} instead of T^{∗}M. Indeed, let Θ stands for the domain of w, i.e.

Θ := (−∞, 0] ∪ ([0, +∞) × [0, 1]) .  (25)

We start from a trivialization ϕ_{x}: T T^{∗}M|_{N(x)} → N(x) × R^{2n} where N(x) is the normal neighborhood of the path x(t) and trivializations ϕ_{w} of w^{∗}T T^{∗}M , ϕ_{v} of w^{∗}T T^{∗}M such that ϕ_{x}(s, t), ϕ_{w}(s, t), ϕ_{v}(s, t) are bounded uniformly in (s, t) in the operator norm. For ρ ≥ ρ_{0} we define a trivialization ϕ_{ρ} of (w♯_{ρ}^{0}v)^{∗}T T^{∗}M such that

ϕ_{ρ}|_{[ \frac{ρ}{2} ,ρ+1]} ≡ ϕ_{x}|_{[ \frac{ρ}{2} ,ρ+1]},
ϕ_{ρ}|_{(−∞, \frac{ρ}{2} ]} ≡ ϕ_{1} ∘ ϕ_{w}|_{(−∞, \frac{ρ}{2} ]},
ϕ_{ρ}|_{[ρ+1,+∞)} ≡ ϕ_{2} ∘ ϕ_{v}|_{[ρ+1,+∞)}

where

ϕ_{1}(s, t) := ϕ_{x} (\frac{ρ}{2}, t) ∘ ϕ_{w}^{−1}(\frac{ρ}{2}, t), ϕ_{2}(s, t) := ϕ_{x}(ρ + 1, t) ∘ ϕ_{v}^{−1}(ρ + 1, t)

are uniformly bounded in (s, t). Thus we obtain the trivialization

ϕ: ⋃_{ρ≥ρ_{0}}(w♯_{ρ}^{0}v)^{∗}T T^{∗}M → [ρ_{0}, +∞) × Θ × R^{2n}

which is uniformly bounded in the operator norm. So the estimates we want to prove stay unchanged when we transfer the settings into trivial framework.

The strategy is the following: near the breaking Hamiltonian x(t) we use the fact that the asymptotic linearized operators are isomorphisms and away from x(t) we use that the pre-glued objects are equal (up to the shifting) to the fixed ones w and u. More precisely, let β: → [0, 1] be a smooth cut-off function equal to 0 for s ∈ (−∞, \frac{1}{2} − ε] ∪ [1 + ε, ∞) and equal to 1 for s ∈ [\frac{1}{2} − \frac{ε}{2}, 1 + \frac{ε}{2}] for fixed, suitably chosen ε > 0. Let β_{n}(s) := β(\frac{s}{ρ_{n}}). From the construction of the pre-gluing map one can obtain

∥D_{χ_{n}}ς_{n}∥_{L^{r}} → 0 ⇒ ∥\frac{∂}{∂s}(β_{n}ς_{n}) + J(∞, ·)\frac{∂β_{n}ς_{n}}{∂t} + A(∞, ·)β_{n}ς_{n}∥_{L^{r}} → 0

(see [4, 6] or [19] for the details). Since \frac{∂}{∂s} + J(∞)\frac{∂}{∂t} + A(∞) is an isomorphism

we see that β_{n}ς_{n} ⟶^{W^{1,r}} 0, so

∥ς_{n}|_{[ \frac{ρn}{2} ,ρ_{n}+1]}∥_{W^{1,r}} → 0.  (26)

Denote by β^{−}(s) := β^{+}(−s), where β^{+} is given in (16) and by β_{τ}^{±}(s) := β^{±}(s + τ), ς_{τ} (s, t) := ς(τ + s, t). Now it follows from (24) and (26) that

∥D_{w}(β_{− \frac{ρn}{2} −1}^{−}ς_{n})∥_{L^{r}} → 0 and ∥D_{v}(β_{ρ_{n}}^{+}(ς_{n})_{2ρ_{n}})∥_{L^{r}} → 0.  (27)

But β_{− \frac{ρn}{2} −1}^{−}ς_{n} ∈ L̃_{w}^{⊥} and β_{ρ_{n}}^{+}(ς_{n})_{2ρ_{n}} ∈ L̃_{u}^{⊥} and for such vectors it holds:

\begin{matrix}∥D_{w}(β_{− \frac{ρn}{2} −1}^{−}ς_{n})∥_{L^{r}} ≥ c_{1}∥β_{− \frac{ρn}{2} −1}^{−}ς_{n}∥_{W^{1,r}} \\ ∥D_{v}(β_{ρ_{n}}^{+}(ς_{n})_{2ρ_{n}})∥_{L^{r}} ≥ c_{2}∥β_{ρ_{n}}^{+}(ς_{n})_{2ρ_{n}}∥_{W^{1,r}} \end{matrix}  (28)

since D_{w}|_{L_{w}^{⊥}} and D_{v}|_{L_{u}^{⊥}} are isomorphisms.

We now conclude:

1 = lim_{n→∞} ∥ς_{n}∥_{W_{ϖn}^{1,r}}

= lim_{n→∞} ∥β_{−ρ_{n}}^{+}ς_{n} + β_{− \frac{ρn}{2} −1}^{−}ς_{n} + (1 − β_{−ρ_{n}}^{+} − β_{− \frac{ρn}{2} −1}^{−})ς_{n}∥_{W^{1,r}}

≤ lim_{n→∞} (∥β_{−ρ_{n}}^{+}ς_{n}∥_{W^{1,r}} + ∥β_{− \frac{ρn}{2} −1}^{−}ς_{n}∥_{W^{1,r}} + ∥(1 − β_{ρ_{n}}^{+} − β_{− \frac{ρn}{2} −1}^{−})ς_{n}∥_{W^{1,r}})

=^{(i)} lim_{n→∞} (∥β_{ρ_{n}}^{+}(ς_{n})_{2ρ_{n}}∥_{W^{1,r}} + ∥β_{− \frac{ρn}{2} −1}^{−}ς_{n}∥_{W^{1,r}})
≤^{(ii)} lim_{n→∞} (\frac{1}{c_{1}} ∥D_{w}(β_{− \frac{ρn}{2} −1}^{−}ς_{n})∥_{L^{r}} + \frac{1}{c_{2}} ∥D_{v}(β_{ρ_{n}}^{+}(ς_{n})_{2ρ_{n}})∥_{L^{r}})

=^{(iii)} 0.

The equality (i) follows from (26), the fact that

supp(1 − β_{−ρ_{n}}^{+} − β_{− \frac{ρn}{2} −1}^{−}) ⊂ [\frac{ρ_{n}}{2} , ρ_{n} + 1]

and the equality ∥β_{−ρ_{n}}^{+}ς_{n}∥_{W^{1,r}} = ∥β_{ρ_{n}}^{+}(ς_{n})_{2ρ_{n}}∥_{W^{1,r}}; (ii) follows from (28) and (iii) follows from (27). We obtain the contradiction so the proof follows.

3.2. The existence of the exact solution. Reducing the problem of finding an exact solution to the following abstract Lemma is standard ingredient in all gluing problems.

Lemma 5. [7, 19] Assume that a smooth map f: E → F between Banach spaces E and G has an expansion

f(ς) = f(0) + Df(0)ς + N(ς)

so that Df(0) has a finite dimensional kernel and a right inverse G and so that for ς, ζ ∈ E it holds:

∥GN(ς) − GN(ζ)∥_{E} ≤ C (∥ς∥_{E} + ∥ζ∥_{E}) ∥ς − ζ∥_{E}

for some constant C. Set ε = \frac{1}{5C}. If ∥Gf(0)∥_{E} ≤ \frac{ε}{2}, then there exist the unique zero point of the map f

x_{0} ∈ B_{ε}(0) ∩ G(F)

such that ∥x_{0}∥_{E} ≤ 2∥Gf(0)∥_{E}.

The proof of Lemma 5 is based on Banach contraction mapping principle and can be found in [19].

From Proposition 4 it follows that there exists a vector bundle isomorphism

D|_{L^{⊥}}: L^{⊥} →^{≅} E^{0,r}

where

L^{⊥} := ⋃_{χ∈K×[ρ_{0},+∞)} L_{χ}^{⊥}

and E^{0,r} is defined below the formula (20). Its fibre-wise restriction is

D_{χ}|_{L_{χ}^{⊥}}: L_{χ}^{⊥} →^{≅} L_{γ}^{r}(p) × L_{u ♯_{ρ}^{0} v}^{r}(y).

Let G_{χ} be the inverse map G_{χ} := (D_{χ}|_{L_{χ}^{⊥}})^{−1}. In order to apply Lemma 5 to our case, we need the estimate of the right inverse. We prove the following

Lemma 6. There exist constant C and a lower parameter bound ρ_{2} ≥ ρ_{0} such that G satisfies the estimate

∥G_{χ}η∥_{W^{1,r}} ≤ C∥η∥_{L^{r}}  (29)

for all χ ∈ K × [ρ_{2}, +∞), η ∈ L_{γ}^{r}(p) × L_{u ♯_{ρ}^{0} v}^{r}(y).

Proof. The estimate (29) is equivalent to

∥ς∥_{W^{1,r}} ≤ C∥D_{χ}ς∥_{L^{r}}.  (30)

Suppose (30) does not hold, then there exist sequences ρ_{n} → ∞, (u_{n}, v_{n}) ∈ K and ς_{n} ∈ L_{χ_{n}}^{⊥} (where χ_{n} = (w_{n}, v_{n}, ρ_{n})) such that

∥ς_{n}∥_{W^{1,r}} = 1, ∥D_{χ_{n}}ς_{n}∥_{L^{r}} → 0.  (31)

The rest of the proof can be reduced to the arguments similar to the ones from the proof of Proposition 4. The main difference is the fact that here we have the sequence (w_{n}, v_{n}) of the broken trajectories and the sequence ρ_{n} → ∞, instead of the fixed one (w, v) and the sequence ρ_{n} → ∞ as it was there. This difficulty can

be solved by assuming, due to the compactness of K that (w_{n}, v_{n}) ⟶^{W^{1,r}} (w, v) ∈ K. Denote by

χ_{m,n} := (w_{n}, v_{n}, ρ_{m}), χ_{n} := (w, v, ρ_{n})
ϖ_{m,n} := w_{n} ♯_{ρ_{m}}^{0}v_{n}, ϖ_{n} := w ♯_{ρ_{n}}^{0}v.

Let U ⊂ T^{∗}M be the open neighborhood of w(Θ) ∪ u(D) (see (25) and (7) and

Φ: T T^{∗}M|_{U} →^{≈} U × R^{2n}  (32)

such that ϖ_{χ_{m,n}}(Θ) ⊂ U for m, n ≥ n_{0}. Define

Φ_{m,n}: ξ ↦ Φ^{−1}(w ♯_{ρ_{m}}^{0}v, Proj_{2} ∘Φ(ξ))

where Proj_{2} denotes the projection to the second component in the right side of (32). We see that Φ_{n,n} establishes the isomorphism between W_{ϖ_{n,n}}^{1,r} and W_{ϖ_{n}}^{1,r} as well as between L_{γ_{n}}^{r} × L_{u_{n} ♯_{ρn}^{0} v_{n}}^{r} and L_{γ}^{r} × L_{u ♯_{ρn}^{0} v}^{r} such that (due to the condition

(w_{n}, v_{n}) ⟶^{W^{1,r}} (w, v)) ϕ_{n,n} → Id_{W_{ϖn}^{1,r}} when n → ∞. So, for given ε > 0 we transform the condition (31) to

1 − ε ≤ ∥(Proj_{L_{χn}^{⊥}} ∘Φ_{n,n})ς_{n}∥_{W^{1,r}} ≤ 1 + ε, ∥D_{χ_{n}}ς_{n}∥_{L^{r}} → 0

for n ≥ n_{0}. The map Proj_{L_{χn}^{⊥}} is well defined due to the fact that L_{χ_{n}}^{⊥} complemented in W_{ϖ_{n}}^{1,r}(p, y) (since its complement is Ker D_{χ} hence finite-dimensional). But (Proj_{L_{χn}^{⊥}} ∘Φ_{n,n})ς_{n} is the sequence of vectors in L_{χ_{n}}^{⊥} so we are in the situation from Proposition 4. The rest of the proof is as there (see [19] for more details).

Now we apply Lemma 5 to our situation. The map f from Lemma 5 is, in our case, the fibre-wise restriction

F_{χ}: W_{ϖ_{χ}}^{1,r}(p, y) → L_{γ}^{r}(p) × L_{u ♯_{ρ}^{0} v}^{r}(y)

of a bundle map (20). The maps Df and N from Lemma 5 are the corresponding derivation terms in the expansion of F_{χ}. So Df is D_{χ}. The map G from Lemma 5 is the map G_{χ} from Lemma 6. The only assumption from Lemma 5 left to check is the assumption about the non-linear term N. From the following three expansions:

F_{χ}(ξ) = F_{χ}(0) + D_{χ}(0)ξ + N_{χ}(0, ξ)
F_{χ}(η) = F_{χ}(0) + D_{χ}(0)η + N_{χ}(0, η)
F_{χ}(ξ) = F_{χ}(η) + D_{χ}(η)(ξ − η) + N_{χ}(η, ξ − η)

we compute

N_{χ}(ξ) − N_{χ}(η) = F_{χ}(ξ) − F_{χ}(η) − D_{χ}(0)(ξ − η)
= (D_{χ}(η) − D_{χ}(0)) (ξ − η) + N_{χ}(η, ξ − η).

Since K is compact, the set

⋃_{χ∈K×[ρ_{0},+∞)} ϖ_{χ}(Θ)  (33)

is relatively compact in T^{∗}M so we have an estimate:

∥N_{χ}(ξ) − N_{χ}(η)∥ ≤ C_{1} (∥ξ∥_{W_{χ}^{1,r}} + ∥η∥_{W_{χ}^{1,r}}) ∥ξ − η∥_{W_{ϖ}^{1,r}}  (34)

where the constant C_{1} depends on the C^{2}− norm of the ∇_{2} exp on a relatively compact set (33). Now the required estimate for GN in Lemma 5 follows from (34) and the fact that G_{χ} from Lemma 6 is bounded.

It follows from Lemma 5 that there exists the unique solution Γ(χ) ∈ B_{ε}(0)∩L_{χ}^{⊥} satisfying F_{χ}(Γ(χ)) = 0. If we set w ♯_{ρ} v := exp_{w ♯_{ρ}^{0} v} Γ(χ) we obtain the exact solution satisfying F (w ♯_{ρ} v) = 0. From the estimates

∥F (ϖ_{χ})∥_{L^{r}} ≤ αe^{−mρ}  (35)

for some α > 0, m > 0 and

∥Γ(χ)∥_{W^{1,r}} ≤ 2C∥F (ϖ_{χ})∥_{L^{r}}  (36)

it follows that

∥Γ(χ)∥_{W^{1,r}} ≤ C(K)e^{−mρ}  (37)

for some C(K) > 0 depending on K. The estimate (36) follows from Lemma 5 with C depending on right inverse G. The estimate (35) can be easily obtained from the construction of the pre-glued trajectory ϖ and the exponential convergence of w, v toward x(t) when s → ±∞ (see [5] for details).

We deduce from (37) that the difference of the approximately glued trajectories w ♯_{ρ}^{0} v in (18) and the exact solution w ♯_{ρ} v tends to zero fast enough when ρ → ∞. Obviously the pre-glued trajectories w ♯_{ρ}^{0} v already converge to the given broken one in the weak sense.

The exact solution of (3) is smooth, this follows from elliptic regularity theory. The only part in Theorem 2 left to prove is the embedding property.

3.3. Embedding property. To show that the gluing map ♯ is an embedding, for ρ ≥ ρ(K) we need to prove that its linearization D♯ is an isomorphism at every point (w, v̂, ρ) and that ♯ is injective.

In section 3.1 we defined pre-gluing for elements from M(p, f; x, H) × M(x, y, H). Let a be fixed regular value of A_{H}. There is an identification

M^{a}(x, y; H) ≅ M̂(x, y, H)  (38)

where

M^{a}(x, y; H) = {v ∈ M(x, y; H) | A_{H}(v(0, ·)) = a}.

Using the inclusion

ι: M^{a}(x, y, H) ↪ M(x, y, H)

we can define gluing for (w, v̂) ∈ M(p, f; x, H) × M̂(x, y, H) as ♯ ∘ (Id, ι) (assuming the identification (38)).

Since K is compact, and regularity is open condition, it is enough to find ρ(w, v̂) for a fixed pair (w, v̂) such that, for ρ ≥ ρ(w, v̂), D ♯(w, v̂, ρ) is an isomorphism. Then we take ρ(K) := max_{(w,v̂)∈K} ρ(w, v̂).

To prove that D ♯ (w, v̂, ρ) is an isomorphism, it is enough to prove that it is injective, since, from indices formulae, we have:

dim (K × [ρ(K), ∞)) = m_{f}(p) − (μ_{H}(x) + \frac{n}{2}) + (μ_{H}(x) − μ_{H}(y) − 1) + 1

= m_{f}(P) − (μ_{H}(y) + \frac{n}{2}) = dim M(p, f; y, H).

Assume it is not true, i.e. that there exists no ρ(w, v̂) such that D ♯ (w, v̂, ρ) is injective for ρ ≥ ρ(w, v̂). Then, there exist sequences ξ_{n} ∈ T_{w}M(p, f; x, H), ζ_{n} ∈ Tv̂M̂(x, y, H), ρ_{n} → ∞ and t_{n} ∈ T_{ρ_{n}}[ρ_{0}, ∞) such that

(ξ_{n}, ζ_{n}, t_{n}) ≠ (0, 0, 0), (ξ_{n}, ζ_{n}, t_{n}) ∈ Ker D ♯(u, v, ρ_{n}).  (39)

Assume, since kernel is a vector subspace, that

∥ξ_{n}∥ + ∥ζ_{n}∥ + |t_{n}| = 1.  (40)

We have

D ♯ (w, v̂, ρ)(ξ, ζ, t) = D ♯_{ρ} (w, v)(ξ, ζ) + D ♯ (w, v) · (ẇ, −v̇) · t,

where first D denotes derivative in (w, v) and the second D derivative in (w, v, ρ). Further:

D ♯_{ρ}(w, v)(ξ, ζ) = ∇_{1} exp(Γ(χ))(ξ ♯_{ρ} ζ) + ∇_{2} exp(Γ(χ)) (DΓ(χ)(ξ, ζ))

and

D ♯ (w, v)(ẇ, −v̇)t = ∇_{1} exp(Γ(χ))(ẇ ♯_{ρ} (−v̇)t)

so from (39) we conclude

0 = D ♯ (ξ_{n}, ζ_{n}, t_{n})

= ∇_{1} exp(Γ(χ_{n}))[(ξ_{n} ♯_{ρ_{n}} ζ_{n}) + ẇ ♯_{ρ_{n}} (−v̇)t_{n}] + ∇_{2} exp(Γ(χ_{n})) (DΓ(χ_{n})(ξ_{n}, ζ_{n}))

i.e.

∇_{1} exp(Γ(χ_{n}))[(ξ_{n} ♯_{ρ_{n}} ζ_{n}) + ẇ ♯_{ρ_{n}} (−v̇)t_{n}] = −∇_{2} exp(Γ(χ_{n})) (DΓ(χ_{n})(ξ_{n}, ζ_{n})) .

But from the identity (∇_{2} exp^{−1} ∘∇_{1} exp)(p, 0) = Id_{T_{p}P} and the exponential decrease ∥Γ(χ_{n})∥, ∥DΓ(χ_{n})∥ ⟶^{n→∞} 0, we conclude

∥(ξ_{n} ♯_{ρ_{n}} ζ_{n}) + ẇ ♯_{ρ_{n}} (−v̇)t_{n}∥_{W^{1,r}} → 0.

So it follows

∥ξ_{n} + ẇt_{n}∥_{W^{1,r}(−∞, \frac{ρn}{2} ]} → 0, ∥ζ_{n} − v̇t_{n}∥_{W^{1,r}[−2ρ_{n},+∞)} → 0.

Since ζ_{n} ∈ Tv̂M̂(x, y, H) and Tv̂M̂(x, y, H) ∩ Rv̇ = {0} we conclude t_{n} → 0, so

∥ξ_{n}∥, ∥ζ_{n}∥ → 0

which is in contradiction with (40). Hence we proved the regularity of D ♯.

The injectivity of ♯ (for ρ large enough) follows from already proved regularity and the compactness of K. We argue by contradiction, so assume that there exist sequences ρ_{n} → ∞,

(w_{n}, v̂_{n}) ≠ (δ_{n}, σ̂_{n})  (41)

such that w_{n} ♯_{ρ_{n}} v̂_{n} = δ_{n} ♯_{ρ_{n}} σ̂_{n}. Since K is compact we can assume that

w_{n} → w, v̂_{n} → v̂, δ_{n} → δ, σ̂_{n} → σ̂.

It follows from exponential decrease of Γ(χ_{n}) = Γ(w_{n}, v̂_{n}, ρ_{n}) and Γ(λ_{n}) = Γ(δ_{n}, σ̂_{n}, ρ_{n}) that in local coordinates:

∥w ♯_{ρ_{n}}^{0} v − δ ♯_{ρ_{n}}^{0} σ∥_{W^{1,r}} ⟶^{n→∞} 0

so w = δ, v̂ = σ̂. Since we already proved that ♯ is local diffeomorphism, we conclude that there exists n_{0} ∈ N such that for all n ≥ n_{0} it holds

(w_{n}, v̂_{n}) = (δ_{n}, σ̂_{n})

which contradicts (41).

We have finished the proof of the part a) of Theorem 2. The proof of the part b) goes analogously.

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(received 10.08.2007, in revised form 20.09.2007)

Matematički fakultet, Studentski trg 16, 11000 Belgrade, Serbia

E-mail: jelenak@matf.bg.ac.yu