  We develop an analytic theory of existence and regularity of surfaces (given
by graphs) arising from the geometric minimization problem
$$\min_{\mathcal{M}}\frac{1}{2}\int_{\mathcal{M}}|\nabla_{\mathcal{M}}H|^2\,dA$$
where $\mathcal{M}$ ranges over all $n$-dimensional manifolds in
$\mathbb{R}^{n+1}$ with prescribed boundary, $\nabla_{\mathcal{M}}H$ is the
tangential gradient along $\mathcal{M}$ of the mean curvature $H$ of
$\mathcal{M}$ and $dA$ is the differential of surface area. The minimizers,
called surfaces of minimum mean curvature variation, are central in
applications of computer-aided design, computer-aided manufacturing and
mechanics. Our main results show the existence of both smooth surfaces and of
variational solutions to the minimization problem together with geometric
regularity results. These are the first analytic results available on the
literature for this problem.
