Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 25I / c

Solution

Put

h (r, θ) = G (r, θ)

h_{rr} = - K h, h (0, θ) = 0, h_{r} (0, θ) = 1.

Inside a geodesic polar coordinate ball,

h > 0

. If

K \leq 0

, then

h_{rr} \geq 0

, so

h_{r} \geq 1

and therefore

h (r, θ) \geq r

h d r d θ

, whence

Area B (p, ε_{0}) = \int_{0}^{2 π} \int_{0}^{ε_{0}} h (r, θ) d r d θ \geq \int_{0}^{2 π} \int_{0}^{ε_{0}} r d r d θ = π ε_{0}^{2} .

Thus nonpositive Gaussian curvature makes such a ball at least as large as the Euclidean disk of the same radius.

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