# What is the Willmore energy of the Earth (or the geoid)?

Wikipedia defines the Willmore energy as:

$$e[{\mathcal{M}}]=\frac{1}{2} \int_{\mathcal{M}} H^2\, \mathrm{d}A,$$

where $H$ stands for the mean curvature of the manifold $\mathcal{M}$.

What is the Willmore energy of the Earth, or the geoid?

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## 1 Answer

Let's idealize the system for a moment :)

Say the earth is a sphere. Its mean curvature is $R^{-1}$ with $R$ the radius. Then the integral becomes

$\int_{\text{spherical surface}} R^{-2} R^2 d\Omega$=surface of a sphere=$4\pi$

as it should be.I guess this can somehow be nicely generalized to a geoid but I don't see how at the moment. Hope it still helps though!

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