Uniform area quantum and finite homology classes for closed tame holomorphic curves

Under [A1], by [F1], write $H_2(X;\mathbb{Z})={\mathbb{Z}}^r\oplus T$, with $T$ finite, and choose the dual coordinate homomorphisms $a_1,\dots ,a_r$ to $\mathbb{R}$. By [F2–F3], choose smooth closed forms ${\eta }_i$ such that ${\int }_u{\eta }_i=a_i(u_{\ast }[C])$. In particular these numbers are integers. This does not assume that the periods of $\omega$ form a discrete subgroup; those periods can be dense. There are only finitely many chosen auxiliary forms.

Tameness and compactness give constants $c>0$ and $M_i<\mathrm{\infty }$ with ${\omega }_x(v,J_{x}v)\ge c\mathrm{\mid }v{\mathrm{\mid }}^2,\mathrm{\mid }{\eta }_i(v,J_{x}v)\mathrm{\mid }\le M_i\mathrm{\mid }v{\mathrm{\mid }}^2\mathrm{.}$ Choose any fixed target metric. The same constants work over a compact tame family: the continuous positive function on its unit tangent sphere has a positive minimum, and each form expression has a finite maximum. Set $C_i=M_i\mathrm{/}c$. For any $J$-holomorphic map, an oriented orthonormal domain frame $(e,je)$ gives $du(je)=Jdu(e)$, whence pointwise $\mathrm{\mid }{u}^{\ast }{\eta }_i(e,je)\mathrm{\mid }\le C_i{u}^{\ast }\omega (e,je)\mathrm{.}$ Integrating gives $\mathrm{\mid }{a}_i(u_{\ast }[C])\mathrm{\mid }\le C_i{\int }_C{u}^{\ast }\omega$, without any restriction on the domain geometry. The same inequality adds over finitely many components.

Choose $h>0$ with $C_{i}h<1$ for every $i$, taking any positive $h$ if $r=0$. If a closed holomorphic map had area $0<A<h$, every integer $a_i$ would have absolute value below one and hence be zero. Its homology class would therefore be torsion. A real closed form evaluates to zero on a torsion cycle: if $m\alpha =0$, then $m⟨[\omega ],\alpha ⟩=0$ in $\mathbb{R}$. Thus its area would be zero, a contradiction. Area zero itself forces $du=0$ pointwise by the positive tame inequality; connectedness then makes the map constant. This proves the uniform domain-independent quantum.

If total area is at most $A$, step 2.1 confines every free coordinate $a_i$ to the finite integer interval $[-C_{i}A,C_{i}A]$. The torsion coordinate lies in the finite set $T$. Their product contains only finitely many integral homology classes. Every nonconstant normalization component has area at least $h$ by step 3.1, and each constant component has zero area. Summing the nonnegative component areas gives the stated component bound. These arguments also cover a disconnected target (it has finitely many components) and the case $r=0$, where no nonconstant closed tame holomorphic map exists. Boundary and inhomogeneous maps lack the closed integral two-cycle used here and have not been included.