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!


**Addendum**. 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.



Measures of the complexity of a metric
I think you would be pretty happy with the Willmore functional for, well, compact orientable
$C^\infty$ surfaces in $\mathbb R^3.$ It is just the integral of the square of the *mean* curvature or
$$ \frac{1}{2 \pi} \int\_{M^2} \; \; H^2 \; dS $$
This quantity is at least 2, and is only equal to 2 for a round sphere. The Willmore Conjecture is that the minimum for an imbedded torus is achieved on the torus (sometimes called the Clifford torus, by the Bryant correspondence) created by revolving a circle of radius 1 with its center at distance $\sqrt 2$ from the axis of revolution. Here the functional has value $ \pi.$ Leon Simon proved that the minimum (a priori the infimum) is achieved. Rob Kusner found some rather earlier references (before Willmore) to this problem. $$ $$ See, for example, "Total Curvature in Riemannian Geometry" by Thomas J. Willmore. $$ $$ I do not expect there would be much trouble making a discrete version of this.
$$ $$
NOTE: sometimes Willmore writes with the $2 \pi$ divisor, sometimes not.
$$ $$
I found a nice wiki page and some pdf's with references and other information, one a schedule for an October 2010 seminar at Oberwolfach. Anyway,
<http://en.wikipedia.org/wiki/Willmore_energy> and 
<http://www.mfo.de/programme/schedule/2010/43b/programme1043b.pdf> and  

<http://www.warwick.ac.uk/~maseq/wmsri.pdf> and 
<http://www.math.ethz.ch/~riviere/papers/riviere-tartar.pdf>  

$$ $$
I was not aware of this, it seems the discrete version of this has been worked out, a fair amount published, including treatment in a book,
"Discrete differential geometry"
 by Alexander I. Bobenko, which can be viewed with google books. I ran google with "discrete willmore functional."


