The mean curvature $H$ of a surface $S$ is given by $$ H = \frac12(\kappa_1 + \kappa_2) $$ where $\kappa_i$ are the principal curvatures, which are the eigenvalues of the shape operator $\text{S}$ over our surface.
My question is this: why divide by 2? As far as I can tell there is no reasonable explanation for why we define $H = \frac12(\kappa_1 + \kappa_2)$, and not simply $H = \kappa_1 + \kappa_2$.
At first I thought maybe it's the (arithmetic) "mean" of the principal curvatures, and the Gaussian curvature analogously is the geometric mean of the principal curvatures, but that's not correct because the Gaussin curvature $K = \kappa_1\kappa_2$, which isn't quite the geometric mean, so why should $H$ be the arithmetic mean?
Every formula I see involving $H$ just involves an unnecessary $2$. For example,
- $$ H = \frac{EN - 2FM + GL}{2(EG - F^2)} $$ where $E, F, G$ are the coefficients of the first fundamental form, and $L, M, N$ are the coefficients of the second fundamental form. Why not just $$ H = \frac{EN - 2FM + GL}{EG - F^2}? $$
- Expressing it as the trace of the shape operator (or the Weingarten map) we see $$ H = \frac12\text{trace}(S) $$ why not just $H = \text{trace}(S)?$
- In the equation relating the three fundamental forms, we see $$ \mathbf{III}-2H\mathbf{II}+K\mathbf{I} = 0 $$ why not just $$ \mathbf{III}-H\mathbf{II}+K\mathbf{I} = 0? $$
- Changing $H$ to $\kappa_1 + \kappa_2$ doesn't affect the theory of minimal surfaces where $H = 0$.
The only compelling evidence for why this might be a reasonable definition is that, for hypersurfaces: $$ H = \frac1n\sum_{i=1}^n \kappa_i $$ according to Spivak, but I don't know much about hypersurfaces so I can't comment on how comparatively useful this definition is.
So what's the deal? Was $H = \kappa_1 + \kappa_2$ originally, but when hypersurfaces were studied the definition $H = \frac1n\sum\kappa_i$ was more natural, and later adopted for the $n = 2$ case as well?