In Topology and Geometry, a curve pseudoholomorphe is an application of a Surface of Riemann, possibly on board, in an almost complex variety satisfying the equations of Cauchy-Riemann. The regularity is imposed by the regularity of the almost complex structure used.

Introduced in 1985 by Mikhaïl Gromov, they play a central role in symplectic Géométrie, and intervene in particular in the definition of the Homologie of Floer.

Definition

In all the article, J indicates an almost complex structure of class C ^K on a differential variety M : in other words, J is a field of operators on the tangent space of M of class C ^K checking J ²=-Id. $\backslash \; Sigma$ is a surface of Riemann, possibly on board; J indicates the associated almost complex structure. By the Theorem of standardization, J is equivalent to the data of a structure in conformity. With the need is introduced the shape of surface D v on surface $\backslash \; Sigma$, which is equivalent specifying metric a riemannienne.

A curve pseudoholomorphe is an application ($\backslash \; Sigma$, J ) $\backslash \; rightarrow$ ( M , J ) checking the identity:

$du\backslash circ\; j=J\backslash circ\; du$.

This identity means exactly that the differential D U : T$\backslash \; Sigma\; \backslash \; rightarrow$T M is a complex fiber morphism vectorial .

The question of the regularity of U in the definition is secondary. The application U is necessarily of class C ^{K -1}.

Energy

In symplectic Geometry, being given a symplectic form $\backslash \; omega$, it is of everyday usage to introduce an almost complex structure J which is $\backslash \; omega$-compatible.

$E\; (U)\; =\; \backslash \; int\_\; \{\backslash \; Sigma\}\; \backslash \; frac\; \{\backslash |\backslash |^2\}\; \{2\}\; dv=\; \backslash \; int\_\; \{\backslash \; Sigma\}\; f^*\; \backslash \; omega$

A first important point is that the finitude of energy authorizes the prolongation of the curves pseudoholomorphes in the points where they are not defined: The proof is based on arguments of analytical nature. For example, under the assumptions of the preceding theorem, a curve pseudoholomorphe C $\backslash \; rightarrow$ ( M , J ) of finished energy is prolonged in a sphere pseudoholomorphe S ²= C $\backslash \; cup\; \backslash \; \{\backslash \; infty\; \backslash \}\; \backslash \; rightarrow$ ( M , J ).

A second important point is a property of quantification of the energy of the spheres pseudoholomorphes:

Theorem - Is a compact symplectic variety ( M , $\backslash \; omega$) and J an almost complex structure $\backslash \; omega$-compatible. Then there exists a finished number of classes of homotopy has in H ₂ ( M , Z ), such as has can be represented by a sphere pseudoholomorphe of energy lower than C .

This result remains valid by replacing J by a compact family of almost complex structures $\backslash \; omega$-compatibles.

Stability

The curves pseudoholomorphes present properties of stability or compactness.

Are a compact variety M , and a continuation J _n of structures almost complex of class C _{$\backslash \; infty$} convergent towards J within the meaning of topology C _{$\backslash \; infty$}. For a surface of Riemann $\backslash \; Sigma$, that is to say a succession of curves pseudoholomorphes U _n: $\backslash \; Sigma\; \backslash \; rightarrow$ ( M , J _n) such as:

for very compact K of $\backslash \; Sigma$. Then, the continuation U _n admits a under-continuation which converges uniformly like all its derivative on very compact of $\backslash \; Sigma$. The limit U is necessarily a curve pseudoholomorphe $\backslash \; Sigma\; \backslash \; rightarrow$ ( M , J ).

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