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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# Measures of the complexity of a metric

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!