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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 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."

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 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 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 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 Oberwohlfach. 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 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 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