Measure of surface quadrature

Question

In the article you can find at http://www.cc.gatech.edu/~turk/my_papers/schange.pdf, precisely at page 2 of the .pdf, there is a functional E which is said to be a measure of the aggregate squared curvature of the surface f(x,y) over the region of interest. I don't know why this functional represent this. Thanks.

Answer 1

The second fundamental form (a matrix which represents extrinsic curvatures) of the graph of $f$ is given by $A = (1 + |\nabla f|)^{-1/2} \nabla^2 f$ where $\nabla^2 f$ is the matrix of second partial derivatives. Thus the functional is

$$ E(f) = \int_\Omega (1+|\nabla f|) |A|^2, $$

so up to the factor of $(1 + |\nabla f|)$ it is just the $L^2$ norm of the second fundamental form. In particular it is zero only when the graph of $f$ is a plane, and will be large when the graph of $f$ is highly curved; so it is a (slightly warped due to the $\nabla f$ term) measurement of the total extrinsic curvature of the surface.