Fairness measures for surfaces are, in general, functionals containing more complicated terms thatn the usual bending energy, and may depend not only on the mean curvature but also on principal curvatures (weighted differently) and their derivatives. Something of the form:
$J(S)=\int_S L(k_1,k_2, \partial k_1,\partial k_2,\ldots)$
A fair surface is, therefore, a surface which is a critical point for one such generalized energy functional, the "easiest" example being minimal surfaces.
There is quite some literature on fair surfaces from the point of view of applications. In particular various constrcutions were developed on how to construct a fair surface from Bezier patches on a grid, trying then to smoothen this discrete approximation in a way that minimizes some fairness measure.
However I was unable to find references for the general theory of minimization of fairness functionals and I wonder if there is any relevant literature on the subject. In only know this PhD thesis http://www.cs.berkeley.edu/~sequin/PAPERS/Theses/Moreton_thesis.pdf
and wonder whether more recent material exists (possibily accesible by a good undergraduate). I am interested in existence results under various boundary conditions, maybe explicit solutions or general proprties of fair surfaces (or relations between different fairness measures).