Pseudo-holomorphic curves: A very quick overview

Abstract This is a review article on pseudo-holomorphic curves which attempts at touching all the main analytical results. The goal is to make a user friendly introduction which is accessible to those without an analytical background. Indeed, the major accomplishment of this review is probably its short length. Nothing in here is original and can be found in more detailed accounts such as [6] and [8]. The exposition of the compactness theorem is somewhat different from that in the standard references and parts of it are imported from harmonic map theory [7], [5]. The references used are listed, but of course any mistake is my own fault.


. Pseudo-holomorphic curves are calibrated
A symplectic form ω has the property that for any compatible J and -plane V we have ω| V ≤ vol V , where vol V is the volume form on V associated with the compatible metric g(·, ·) = ω(·, J·). Moreover, equality is achieved if and only if V is complex. This property, known as Wirtinger's inequality, is enough to show that pseudoholomorphic curves minimize, not only E, but also the area functional in their homology class. Indeed, let u : Σ → M be a pseudo-holomorphic curve andΣ ⊂ M a curve homologous to u(Σ). Then, there is a cobordism N ⊂ M fromΣ to u(Σ) and as ω is closed we nd that = N dω = Σ ω − u(Σ) ω. Using this we compute vol u(Σ) = Area(u(Σ)), which shows that u(Σ) minimizes area in its homology class. We shall state this here for completeness.

Proposition 2 (Area minimization property). Let (M, ω, J) be a symplectic manifold equipped with a compat-
ible almost complex structure and u : Σ → M a pseudo-holomorphic curve¹. Then, u(Σ) minimizes area in its homology class.
As a closing remark, we simply summarize the conclusions we arrived at in two sentences. Pseudoholomorphic curves, are a very special kind of harmonic maps with their image being very special minimal surfaces. They minimize both functionals Area and E, having in fact Area(u(Σ)) = E(u).

Consequences of the minimal surface point of view . The monotonicity formula
A key property of holomorphic curves which is a shadow of them being minimal surfaces is the monotonicity formula. Below we give a "calibrated" proof of this property which the author learned in a talk from Simon Donaldson.

Proposition 3 (Monotonicity)
. Let (Σ, j) be a Riemann surface with a compatible metric g Σ and (M, ω, J) a symplectic manifold u : Σ → M equipped with an almost complex structure. Then, there is R > such that: for allx ∈ u(Σ) and p-holomorphic map u : Σ → M, the quantity is a non-decreasing function of r ∈ ( , R).
Proof for the at metric on M = C n . As u is a p-holomorphic map we may choose g Σ to be the induced metric g| Σ . Furthermore, we may as well suppose with no loss of generality that u(x) = in which case we shall write Br := Br( ). Then, it follows from the energy identity that |du| dvol Σ = u * ω and From this, and writing ω = dθ with |θp| ≤ |p| we nd and so it will be enough to prove that To do this recall that for ρ(·) = | · |, we have L ρ∂ρ ω = ω and using |ι ∂ρ ω| = we have as we wanted to prove.

Remark 3 (Minimal surfaces point of view and non-squeezing).
In the theory of minimal surfaces, the quantity 2.1 is more usually written as r Area(u(Σ) ∩ Br(x)).
Then, the statement is that this quantity is a non-decreasing function of r ∈ ( , R). In particular, the limit of this quantity as r → exists and letting c be this limit we nd that This property was one of the key ingredients used by Gromov in his proof of the nonsequeezing theorem which we shall state and prove in 7. The main idea being that if a symplectic embedding i : B n R → C n r = D r × R n− was to exist for R > r, then the radius R-ball would symplectically t into S r+δ × T n− where r + δ < R. One then argues that it is possible to nd a pseudo-holomorphic sphere C passing through i( ) with image in the class [S r+δ × { }] which then has area (and energy) π(r + ε) . However, then i − (C) is a minimal surface passing through the origin which implies that its area is at least that of the disk through the origin, i.e. πR > π(r + δ) , leading to a contradiction. Of course, the details of the argument are more involved than in this short sketch.
They key point missing resides in proving that such a pseudo-holomorphic curve does exist.

. Removable singularities
The monotonicity property and the minimal surface viewpoint may also be used in proving a removable singularity theorem. Here we state the most general result which asserts that the extended map is smooth, but only prove continuity (the proof here follows that in [6]). Proof of continuity. Suppose not, i.e. that u does not continuously extend over { }, then it has di erent accumulation points as z → , say p ≠ p ∈ M. Let δ < dist(p , p )/ , we shall rst prove that both u − (B δ (p )) and u − (B δ (p )) have a nite number of connected components. Indeed, any such component is a minimal surface passing arbitrarily close to the point p ∈ M. Thus, by the monotonicity formula for some constant c only depending on the target geometry. In particular, as C ⊂ Σ and E(u) = Area(u(Σ)) is nite by hypothesis, we nd that there can be at most a nite number of such components. Thus, u − (B δ (p )) and u − (B δ (p )) have a nite number of components and so there is a connected component of each of these containing as an accumulation point. From this we conclude that for su ciently small r > the circles ∂Dr intersect both these components and so u(∂Dr) meets both B δ (p ) and B δ (p ). As dist(p , p ) > δ we have Length(u(∂Dr)) ≥ δ for all small r. Furthermore, as u is holomorphic

Intuition for main compactness result
This section explain the main intuition behind the failure of a family of pseudo-holomorphic curves to converge to another one in a nice manner. Our exposition is motivated by that in [3].

. Comparison with geodesics
Geodesics in a Riemannian manifold (M, g) locally minimize length and are critical points of the energy functional The same compactness phenomenon does not generalize to dimensions greater than one. Indeed we can suspect that to be the case already from the failure of the embedding L → C in dimension as the next example shows.

Example 1 (Failure of borderline Sobolev embedding). For r =
x + y , the function f (r) = log log r − in D ⊂ R has square integrable derivative, i.e. energy or L -norm, but is unbounded and so not in C . Indeed, From the point of view of a family of pseudo-holomorphic maps {u k } k∈N , with u k : Σ → M, the failure of this Sobolev embedding suggests that an energy bound may not be enough to guarantee the existence of a uniformly convergent subsequence. That is indeed the case as we shall see in the next example.

. Bubbling example
Let Σ = CP = M with the Fubini-Study structures which in homogeneous coordinates [ : z] ∈ Σ and [ : w] ∈ M we write as .
Then, we letũ∞ : C → C be a rational map which when written as a quotient of two polynomials p(z) q(z) with no common factors with q having degree n. We will further suppose that u∞(∞) = (this amounts to assuming deg(q) > deg(p)) and that q( ) ≠ . This extends to a rational map which we interpret as u∞ : Σ → M. Then, E(u∞) = πn. Then, we consider the maps which also satisfy u k (∞) = . Our assumptions on p and q imply that these maps u k have degree n+ , i.e. they have energy π(n + ). It is easy to see that, away from the point = [ : ], the maps u k uniformly converge with all derivatives to u∞. We see here the following teo phenomena: However, notice that in the example above u∞ has degree n and so energy This raises the question: where did the extra π of energy go? The answer is roughly that as k → +∞ it got concentrated at and escaped (or bubbled o ) through there. To understand this we compute the energy density at z = . Indeed, using the auxiliar map v with the dots denoting lower order terms. To compute this we nd v * k dw = − kz dz and so which shows that indeed the energy is concentrating at z = . In order to recover the π of energy that is there bubbling o , we rescale at z = with speed |du k ( )|. Then, we de ne the blow up sequence which in compact subsets of C ∼ = T Σ uniformly converges to the map z → /z. This can be extended to the which accounts for the extra π of energy which we had initially lost. Such map is known as the bubble at .
We summarize this as a third phenomenon: (c) Bubbling at the blow up locus, meaning that after appropriately rescaling at the point ∈ S, the rescaled maps u ,k converge to a map u ,∞ : CP → M, the bubble, which accounts for the energy loss at the blow up locus.

Compactness
This section explains the main aspects behind the standard compactness result for pseudo-holomorphic maps. Our approach is based on ideas from harmonic map theory [5,7].  The proof of this theorem will take over the remaining of this section where we shall prove (a) and (b), and the next section where we shall prove (c).

. Main analytical estimate
The following local energy estimate is the key ingredient in identifying the blow up set.
Let u : Σ → M be a pseudo-holomorphic curve, e = |du| its energy density and x ∈ Σ. Then, for any positive r ≤ r The real proof of this result is given in the Appendix. It relies on a second order inequality for the energy density which we shall show satis es ∆e e . This should be interpreted in the following way. Whenever a function e satis es a di erential inequality of the kind ∆e e one can write a mean value inequality which generalizes the standard mean value property of harmonic functions. However, if the right hand side is to be quadratic in e rather than linear, the same inequality need not hold. For that to be the case, we need e to be su ciently small, which is precisely what the condition Dr(x) e ≤ ε < ε guarantees. Here we shall give one other di erent fake argument with the hope of further elucidating why such a result is supposed to elucidate why a result like this should hold.
Fake proof of the ε-regularity. We study the somewhat simpler equation The next result shows that S is a nite set of points. In fact, we shall give an upper bound on its cardinality by bounding its zero dimensional Hausdor measure. Lemma 1. In the setting above, the blow up set satis es H (S) ≤ ε − [ω], β .
Proof. Then, let δ > and consider the cover of S given by x∈S D δ (x). By Vitali's covering lemma we can nd a countable subcollection of disks {D δ (x i )} i∈I which are pairwise disjoint but such that i∈I D δ (x i ) covers S. Then, the counting measure H of S can be bounded via where we have used the fact that the disks D δ (x i ) are disjoint. As the bound above is independent of δ > we nd that [ω], β ε , as claimed in the statement.

Remark 5.
Recall that E(u k ) = [ω], β is the energy of each member of the family. As, the estimate above shows that there can at most [ω],β ε bubbles, we may interpret ε as a lower bound on the minimum energy of each bubble.

. Convergence away from the blow up set
Away from the set S de ned in 4.1, the ε-regularity result in proposition 5 is the key step in guaranteeing the smooth convergence of the sequence u k → u∞ in Σ\S. We state this as follows.
Then, up to passing to a subsequence {u , and proceed as follows Furthermore, completing the square using bound on e k from its L p+ and L p+ bounds. It is also possible to obtain analogous estimates on all higher derivatives of e k by di erentiating the pseudo-holomorphic curve equation and nding inequalities for ∆|∇ l du k | instead of ∆e k . All this gives su ciently strong bounds on the maps u k to guarantee that all their derivatives in K are equicontinuous. At this point, a similar application of Ascoli-Arzelá type theorems, in the form of compact Sobolev embeddings, as that we used for geodesics in section 3.1 yields the stated convergence of the maps u k .

Remark 6. The proof of parts (a) in Theorem 1 is now complete. Part (b) follows from the removable singularities
criteria stated in Proposition 4. The only subtlety in applying it is that we must guarantee that the map u∞ is pseudo-holomorphic. That follows from the fact that the u k are converging to it with all derivatives.

Bubbling
The following result proves part (c) of Theorem 1 by showing that at each point of the blow up set S there is a pseudo-holomorphic sphere bubbling o . This is usually called the "inner bubble" at x. In general there may be a bubble tree⁴ forming and we need to proceed in a more careful manner to see the di erent bubbles.
We conclude that the sequence ux k : D r/δ k ( ) → M, if converging, cannot converge to a constant map. We must now prove that indeed this sequence converges. For this we must put a further restriction on the points x k which selects the "inner bubble". We choose x k not only imposing that δ k = |du k (x k )| − +∞ but we actually choose an x k at which the maximum of |du k | is achieved in Dr(x), i.e. Its energy can be bounded by the fact that the energy of each element in the sequence satis es Meaning bubbles within the bubbles within bubbles and so on. 5 We regard these as being de ned in disks of radius δ − k r inside C ∼ = Tx k Σ. Notice in particular that the metrics gx k geometrically converges to the Euclidean one in compact subsets of C ∼ = Tx k Σ.
This gives a k independent upper bound for the energy of each ux k . Thus, the mapũx : CP → M given bỹ ux(z) = ux( /z) has nite energy and by removal of singularities stated in proposition 4 we nd that this extends to a smooth pseudo-holomorphic mapũx : CP → M. Given that the map z → /z is a biholomorphism we nd that also ux extends to a pseudo-holomorphic sphere. (Noncompact domain curves). When Σ is a strip, as for example in Lagrangian Floer homology, bubbling at boundary points happens in codimension 1. When Σ is a cylinder, as in Hamiltonian Floer homology, bubbling can happen along the ends.

Transversality
As in the introduction, in this section we view the moduli space of pseudo-holomorphic curves as the zero locus of a section∂ J of the bundle T → C ∞ (Σ, M), whose ber at u is Tu = C ∞ (Σ, Λ , Σ ⊗ u * TM). Now, let u be a pseudo-holomorphic map and di erentiate∂ J at u, this gives a map du∂ J : and projecting du∂ J on the second we obtain a map Du∂ given by where in the two last equalities we respectively used the Cauchy-Riemman equation Ju du = du•j, and J = − to compute that Ju dJu + dJu Ju = .
In order to think of Du∂ J of a smooth map between Banach manifolds we must topologize the relevant spaces with Banach space norms. Ideally we would like to work with smooth maps, however the C ∞ -topology is not associated with a norm⁶ and instead we x k ∈ N and p > so that kp > and regard Du∂ J as a map A word must be said about the reason for having kp > . The main reasons for doing this are: First, in this range the Rellich-Kondrachov compactness theorem yields a compact embedding L p k → C . Secondly, in this range the L p k spaces form an algebra which is useful in dealing with the non-linearities. We shall now sketch the proof the following result. Proof. First we prove that the operator Ju(dJu)(·) : L p k → L p k− is compact under the stated hypothesis (this is where the condition that l ≥ k shows up). As the Cauchy-Riemann operator∂ Ju is elliptic, it is Fredholm between the stated function spaces. Having in mind that Du∂ J =∂ Ju − Ju(dJu) and that the Fredholm index is invariant by compact perturbation we conclude that Du∂ J is Fredholm of index index(Du∂ J ) = index(∂ Ju ) which one can compute using the Hirzebruch-Riemann-Roch formula. The condition that u be somewhere injective implies that the map u does not factor through some another curve multiply (branched) covered by Σ.

Main example of application: The nonsqueezing theorem
In this section we will give a self contained proof of Gromov's nonsequeezing theorem by simply making use of the results we have so far presented. For the original proof whose argument we follow see the [4]. There are also several other very good expositions of the nonsqeezing theorem containing all the details, see for example [6] and [8]. Before stating the result we introduce the main characters, the symplectic r-ball and the symplectic R-cylinder. These are are which is the radius r ball equipped with the induced canonical symplectic structure ω = n i= dx i ∧ dy i and which is the radius R cyclinder, again equipped with the induced canonical symplectic structure. Now, we state the theorem.

Theorem 3 (Gromov's nonsqueezing).
There is a symplectic embedding if and only if r ≤ R.
One direction in proving this theorem is quite straightforward as, for r ≤ R, the standard inclusion B n r → C n R is symplectic. Hence, we need only prove the direction asserting that if a symplectic embedding i : B n r → C n R exists then r ≤ R. Starting from the existence of such a symplectic embedding i, the image i(B n ) is bounded in C n R and so, for any δ > we may compactify it inside where T n− is a su ciently large torus and S R+δ a the -sphere of radius R + δ. Then, we equip M with an almost complex structure J which in the image of i agrees with the push-forward of the standard one on B n r . Such a pseudo-holomorphic curve has image C ⊂ M with Area(C) = E(u) = π(R + δ) . On the other hand, pulling C ∩ i(B n r ) back to the ball we nd that as the minimal area minimal surface passing through the origin is the disk of area πr . Hence, πr ≤ π(R + δ) for all δ > nishing the proof of the nonsqueezing theorem.

. Acknowledgments
The author is grateful for the comments and suggestions of an anonymous referee which helped him improve the exposition.

A Proof of the Weitzenböck formula and ε-regularity
This section follows the analogous results in [6] apart from minor rearrangements in the arguments and their exposition.