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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.

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You might want to look into something called discrete differential geometry, used mainly for computer graphics. It doesn't provide a formula, but it could help you measure your surface. In general, something like bregman divergence from a sphere would be good, I think. (Though I don't know exactly how you'd measure it). – DoubleJay Jul 19 '10 at 19:47
Are you looking for an intrinsic invariant or one that might depend on the embedding into $R^3$? – Deane Yang Jul 19 '10 at 20:09
@Deane: Good question! I think I want intrinsic, but I'm not really certain. Sorry to be so vague, but this is at an exploratory stage. – Joseph O'Rourke Jul 19 '10 at 20:14
@DoubleJay: Thanks for mentioning the "Bregman divergence," new to me. – Joseph O'Rourke Jul 19 '10 at 20:17
Here is the dude: – Will Jagy Jul 20 '10 at 3:39
up vote 8 down vote accepted

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, and and and
$$ $$ 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."

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@Will: Ah, very nice! This may be quite useful. Thanks! – Joseph O'Rourke Jul 19 '10 at 20:18
I see that this integral is invariant under conformal maps, a useful property. – Joseph O'Rourke Jul 19 '10 at 22:43
Embarrassingly, I actually own a (signed!) copy of Bobenko's book, and did not recognize the usefulness of his coverage of the Willmore flow! – Joseph O'Rourke Jul 20 '10 at 12:21

I don't think that the Willmore energy is a good measure for the varying of the metric. First of all it is an extrinsic measure, so it measures how the surface lies in the space (In that situation it is, of course, a very good measure). Moreover, the Willmore functional is conformally invariant, so you can apply conformal trnasformations on R3 (better S3) which will change the surface metric in a dramatically way (in general), but it does not change the Willmore functional.

Instead, I would say the best mertic is the one of constant curvature. Of course, there is a energy functional on metrics which has as minimizers the metrics of constant curvature. Take that one.

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Since your metrics embed into R3, you could use the surface area (taken after normalizing the volume) for sufficiently well-behaved metrics. This has the advantage of being easy to calculate in many cases.

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@Robin: Yes, this is one of the "ad hoc" measures I thought of. My concern is that two surfaces that differ in intuitive "complexity" might have the same normalized area. Which raises another question: What do all those surfaces with a given normalized area look like? – Joseph O'Rourke Jul 19 '10 at 20:46
Well, that depends on what intuitive notion of "complexity" you're using. Under the surface-area definition, for example, a highly eccentric ellipsoid will look the same as a sphere with very shallow but convoluted wrinkles, since you're not taking variation in "diameter" into account. – Robin Saunders Jul 19 '10 at 21:06

Another thought is to use the Gromov-Hausdorff metric between metric spaces, where one of the spaces could be the intrinsic metric on the sphere.–Hausdorff_convergence

Also see this paper by Memoli:

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