Nonharmonic exact dbar form

Translation by $m+in$ preserves the Euclidean metric, $d\bar{z}$, and $\sin (2\pi x)$, so these descend to the square torus. The lattice translation action is a covering-space action: disks of radius less than $1\mathrm{/}2$ have disjoint nontrivial lattice translates, so [F4] gives a covering quotient and holomorphic charts with translation transitions. The quotient map is open because the preimage of the image of an open set is the union of its translates. For inequivalent $z,w$, the distance from $z-w$ to the square lattice is positive: only finitely many lattice points lie in any bounded disk, and none equals $z-w$. Small disks about $z,w$ therefore project to disjoint neighborhoods, proving Hausdorffness. Images of a countable base of plane disks form a countable base of the quotient; it is nonempty and connected as a continuous image of $\mathbb{C}$. The image of the closed unit square covers the quotient and is compact, so these charts make it a compact Riemann surface. By [F1], ${\mathrm{\partial }}_{\bar{z}}\sin (2\pi x)=\pi \cos (2\pi x)$; this coefficient equals $\pi$ at $z=0$. Thus $\bar{\mathrm{\partial }}f$ is a nonzero smooth exact form.

If a smooth exact form $a={\bar{\mathrm{\partial }}}_{E}b$ on any compact Riemann surface with the stated metrics is harmonic, [F2] gives ${\bar{D}}^{\ast }a=0$ and $\bar{D}b=a$. The first-variable-linear adjoint identity yields $\mathrm{\parallel }a{\mathrm{\parallel }}^2=⟨\bar{D}b,a⟩=⟨b,{\bar{D}}^{\ast }a⟩=0$, so $a=0$. Applying this to step 1.1 proves that its exact form is not harmonic. The harmonic and exact summands therefore intersect only at zero, as also expressed by [F3]. Full AC is inherited through those operator and Hodge interfaces; the calculation selects no new family.