I am seeking a measure of the "complexity" of a surface $S$,
a quantity that reflects how widely the metric varies from spot to
spot.  I am primarily interested in surfaces topologically
equivalent to a sphere in $\mathbb{R}^3$, so measures that rely
on the genus are not useful.
Ideally the measure would achieve its minimum for a (round) sphere,
would be larger but still small for closed convex surfaces,
and large for surfaces with steep mountains and plummeting valleys.
Ultimately I need to discretize the measure, but I would like to
understand what are the alternatives for smooth metrics.
I can concoct reasonable ad hoc measures, but I'd prefer
to start from a more principled foundation.

From its name, the *entropy of a Riemannian manifold* sounds like it might be
appropriate, but I have only a tenuous grasp of this concept,
so it is unclear to me if this aligns with my goals.
I've also looked at the *systolic ratio* and several other geodesic-based
concepts, but none seem to capture what I want.
I'd appreciate pointers to concepts in this general intellectual neighborhood.
Thanks!

<b>Addendum</b>.  Thanks for the useful suggestions: normalized surface area, Bregman divergence, Gromov-Hausdorff metric, Willmore energy.  My question was too vague to permit a definitive answer,
but I'll accept Will Jagy's suggestions on the Willmore energy, which taught me much.