Pseudoholomorphic curve

For almost complex manifolds $(\Sigma ,j)$ and $(X,J)$, a smooth map $u:\Sigma \to X$ is J-holomorphic when $du\circ j=J\circ du$, equivalently ${\bar{\partial }}_{J}u=0$.

A J-holomorphic curve on a punctured disc with finite energy and image contained in a compact target extends smoothly across the puncture when the target has a smooth tame almost complex structure and compatible bounded geometry. Local monotonicity controls escape from small target balls, and the elliptic small-energy estimate followed by elliptic regularity gives the extension. This is the step turning a finite-energy plane obtained by rescaling into a J-holomorphic sphere.

A J-holomorphic curve is simple if it does not factor through a holomorphic branched covering of degree greater than one. For a nonconstant curve this is equivalent to having a point where its derivative is nonzero and its image has exactly one preimage.

A simple J-holomorphic sphere is a J-holomorphic sphere that is a simple J-holomorphic curve. Its automorphism group as a parametrized map is trivial, because any automorphism fixing the map fixes its locally injective points.

The singular-curve adjunction formula in a symplectic manifold of real dimension four gives $c_1(A)=A^2+2-2\delta$ for a simple J-holomorphic sphere, with a nonnegative singularity count $\delta$. Hence the displayed inequality holds. This is the version needed when the sphere need not be embedded.

A J-holomorphic sphere is a J-holomorphic curve with domain the complex projective line.

For a compact symplectic manifold, smooth convergence of tame almost complex structures and a uniform area bound give a subsequence of J-holomorphic spheres converging, after reparametrization, to a finite tree of J-holomorphic spheres, with convergence smooth away from finitely many bubbling points. The total homology class and symplectic area are preserved. Constant components may be added for stability. In a spherical class which cannot split into two positive-area spherical classes, there is only one nonconstant component.

In a product symplectic manifold $S^2(a)\times T^{2n-2}$, every compatible almost complex structure admits a J-holomorphic sphere in class $F$ through each point. Here $S^2(a)$ has area $a$, and the torus carries a constant symplectic form. All spherical classes are integer multiples of $F$, so a curve in class $F$ cannot split into positive-area nonconstant spheres. For the product complex structure the fibre through a point is unique and regular. Its one-point count modulo two is one; generic transversality and the compact one-dimensional parameterized moduli spaces preserve this count. Approximation and Gromov compactness for spheres remove the genericity requirement. This gives the area-$a$ curve used in the proof of the Gromov non-squeezing theorem.

For a regular nonconstant J-holomorphic sphere in a $2n$-dimensional target, the parametrized Fredholm index is $2n+2c_1(A)$. Removing the six-dimensional Möbius transformation reparametrization group gives the displayed local moduli dimension. The quotient is a manifold for a simple J-holomorphic sphere and may be an orbifold for a multiple cover with finite stabilizer. In dimension four the formula is $2c_1(A)-2$.

This first-order operator vanishes exactly on J-holomorphic curves. Linearization at a map $u$ gives a Fredholm operator between suitable Sobolev spaces of sections. At a constant map it is the Dolbeault operator on the trivial pulled-back tangent bundle.

A real-linear Cauchy–Riemann operator on a complex rank-$n$ vector bundle over a closed genus-$g$ Riemann surface is a Fredholm operator between the usual Sobolev spaces and has the displayed index. Zero-order real-linear terms do not change this index, since they form a continuous deformation of Fredholm operators. For the bundle pulled back by a sphere, subtracting the six-dimensional reparametrization group gives the dimension formula for regular J-holomorphic spheres.

A J-holomorphic curve is regular when the linearized Cauchy–Riemann operator is surjective. Regularity only for simple curves is weaker than regularity for all maps: multiple covers of an exceptional sphere can fail regularity even when that simple sphere is regular.

For a residual set of smooth compatible almost complex structures, all simple J-holomorphic spheres in a specified countable set of spherical classes are regular. The corresponding evaluation maps at marked points are transverse to generic incidence conditions; generic paths give parameterized transversality. The universal linearization can vary the target structure near an injective point of the curve, which removes its cokernel. This assertion concerns simple curves and does not ensure regularity of arbitrary multiple covers.

If every nonconstant J-holomorphic sphere under consideration is regular, no such sphere in a four-dimensional target has class $A$ with $A^2\le -2$. For a simple J-holomorphic sphere, the adjunction inequality for a simple J-holomorphic sphere makes the predicted moduli dimension negative. A degree-$m$ multiple cover of class $B$ has $B^2\le -1$ and $c_1(B)\le 1$, so its predicted real dimension is at most $2m-2$. The rational covering maps of the projective line already form a family of dimension $4m-4>2m-2$, contradicting regularity. The full regularity hypothesis matters: multiple covers of exceptional minus-one spheres violate the conclusion under a simple-only regularity assumption.

A Lagrangian boundary condition for a J-holomorphic curve requires specified boundary components of the source to map into specified Lagrangian submanifolds. For the strip, this is $u(s,0)\in L_0$ and $u(s,1)\in L_1$. For a compatible almost complex structure these target submanifolds are totally real. The condition accompanies the interior equation $u_s+J{u}_t=0$.

For compatible metrics on a Riemann surface and symplectic target,Thus a J-holomorphic curve minimizes energy among maps in its homology class.

On a compact region with controlled compatible almost complex structure, there are $c,r_0>0$ such that a nonconstant J-holomorphic curve through the center of a ball of radius $r<r_0$, with boundary outside the ball, has symplectic area at least $c{r}^2$. For the standard complex structure the sharp local constant is $\pi$.