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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."

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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 proposal schedule for a workshop 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

4 spell Oberwolfach. Go on, I dare you.

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 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 proposal for a workshop at OberwohlfachOberwolfach. 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

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