For a <conformal rescaling of a Riemannian metric> $h=\Omega(dx^2+dy^2)$, the Taubes equation in flat coordinates is
$$
\Delta v+\Omega(1-e^v)=4\pi\sum_rN_r\delta^{(2)}(z-z_r).
$$
The first solution $u$ has $\Omega=1$ and vortex number $N$. For $\widetilde g=e^ug$, the second solution therefore obeys
$$
\Delta\widetilde u+e^u(1-e^{\widetilde u})
=4\pi\sum_s\widetilde N_s\delta^{(2)}(z-\widetilde z_s),
$$
where we used $\Delta_{\widetilde g}=e^{-u}\Delta_g$ and converted the covariant delta distribution to the flat coordinate measure. Adding the equation for $u$ gives
$$
\Delta(u+\widetilde u)+1-e^{u+\widetilde u}
=4\pi\left[
\sum_rN_r\delta^{(2)}(z-z_r)
+\sum_s\widetilde N_s\delta^{(2)}(z-\widetilde z_s)
\right].
$$
Thus $u+\widetilde u$ is again a flat-metric <Taubes equation> solution. Its vortex divisor is the union of the two divisors, with multiplicities added at coincident zeros, and its total vortex number is
$$
\boxed{N_{\rm total}=N+\widetilde N}.
$$
This is <vortex composition by conformal rescaling>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 313 / 2 / d / Solution

{title2=$\widetilde g=e^{2\omega}g$}

A conformal rescaling multiplies a Riemannian metric by a positive scalar function. It preserves angles while changing lengths and volumes; in two dimensions, $\widetilde g=e^{2\omega}g$ gives $\Delta_{\widetilde g}=e^{-2\omega}\Delta_g$ on scalar functions.


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 Incoming links: Conformal rescaling of a Riemannian metric