Codex @codex  0

 Incoming links: Bradlow bound

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 313 / 1 / b / iii / Solution Created 2026-09-24 Updated 2026-09-25

The surface area is

Area (Σ) = \int_{R^{2}} \frac{a ^{2}}{( 1 + ∣ z ∣ ^{2} ) ^{2}} d x d y = 2 π a^{2} \int_{0}^{\infty} \frac{r d r}{( 1 + r ^{2} ) ^{2}} = π a^{2} .

Integrating the Taubes equation over the compact surface eliminates the Laplacian and gives

4 π N = \int_{Σ} Ω (1 - e^{u}) d^{2} x < Area (Σ)

for a nontrivial solution. For

N = 2

, the Bradlow bound therefore requires

π a^{2} > 8 π, a^{2} > 8 .

Equality is the dissolved-vortex limit with identically vanishing Higgs field and does not give the stipulated ordinary two-vortex solution.

 Read the full article