Symplectic area (${\int }_{\Sigma }\omega$)

The symplectic area of a compact symplectic surface $(\Sigma ,\omega )$ is ${\int }_{\Sigma }\omega$, using the orientation determined by the symplectic form.

Two closed, connected symplectic surfaces are symplectomorphic exactly when their genus of a surface and total symplectic area agree. Necessity follows from invariance of topology and the integral of the pulled-back symplectic form.

For sufficiency, the classification of compact surfaces supplies an orientation-preserving diffeomorphism between surfaces of the same genus. Pull both forms back to one surface. Their equal integrals make their classes in $H^2(\Sigma ;\mathbb{R})$ equal. The interpolation ${\omega }_t=(1-t){\omega }_0+t{\omega }_1$ remains positive, since in two dimensions the two positive forms are pointwise positive multiples of each other. Moser theorem now gives an isotopy identifying them.^[ref]

The boundaryless hypothesis matters. A disk and an annulus can have the same genus and the same total area, but different numbers of boundary components prevent even a diffeomorphism. Boundary versions of the classification require the corresponding topology and boundary conditions.

A symplectomorphism of a closed symplectic surface carries the complementary components of a separating simple closed curve to those of its image, preserving their unordered symplectic areas. On the unit sphere a latitude at height $h$ has complementary areas $2\pi (1-h)$ and $2\pi (1+h)$. Thus the equator cannot be sent to any nonequatorial latitude.