  Current quadratic smoothness energies for curved surfaces either exhibit
distortions near the boundary due to zero Neumann boundary conditions, or they
do not correctly account for intrinsic curvature, which leads to
unnatural-looking behavior away from the boundary. This leads to an unfortunate
trade-off: one can either have natural behavior in the interior, or a
distortion-free result at the boundary, but not both. We introduce a
generalized Hessian energy for curved surfaces, expressed in terms of the
covariant one-form Dirichlet energy, the Gaussian curvature, and the exterior
derivative. Energy minimizers solve the Laplace-Beltrami biharmonic equation,
correctly accounting for intrinsic curvature, leading to natural-looking
isolines. On the boundary, minimizers are as-linear-as-possible, which reduces
the distortion of isolines at the boundary. We discretize the covariant
one-form Dirichlet energy using Crouzeix-Raviart finite elements, arriving at a
discrete formulation of the Hessian energy for applications on curved surfaces.
We observe convergence of the discretization in our experiments.
