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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/ii/paper-4/24h/iii/solution

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Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 4 / 24H / iii / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-14

Let

ζ = u + i v

. Direct differentiation gives

φ_{1} = 1 - ζ^{2}, φ_{2} = i (1 + ζ^{2}), φ_{3} = 2 ζ .

(1)

These are holomorphic polynomials and their squared sum is

(1 - ζ^{2})^{2} - (1 + ζ^{2})^{2} + 4 ζ^{2} = 0.

(2)

Moreover direct calculation of the metric gives

E = G = (1 + u^{2} + v^{2})^{2} > 0

and

F = 0

, so the map is genuinely an immersion everywhere. The preceding characterizations prove that it is an isothermal parametrization of a minimal surface, the Enneper surface.

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