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

For a conformal rescaling of a Riemannian metric $h=\Omega (d{x}^2+d{y}^2)$, the Taubes equation in flat coordinates isThe first solution $u$ has $\Omega =1$ and vortex number $N$. For $\tilde{g}=e^{u}g$, the second solution therefore obeyswhere we used ${\Delta }_{\tilde{g}}=e^{-u}{\Delta }_g$ and converted the covariant delta distribution to the flat coordinate measure. Adding the equation for $u$ givesThus $u+\tilde{u}$ 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 isThis is vortex composition by conformal rescaling.