We show that integral curvature energies on surfaces of the type $E_0(M) := \int_M f(x,n_M(x),D n_M(x))\,d\mathcal{H}^2(x)$ have discrete versions for triangular complexes, where the shape operator $D n_M$ is replaced by the piecewise gradient of a piecewise affine edge director field. We combine an ansatz-free asymptotic lower bound for any uniform approximation of a surface with triangular complexes and a recovery sequence consisting of any regular triangulation of the limit sequence and an almost optimal choice of edge director.

翻译：我们证明，对于形如 $E_0(M) := \int_M f(x,n_M(x),D n_M(x))\,d\mathcal{H}^2(x)$ 的曲面上的积分曲率能量，在三角复形上存在离散版本，其中形状算子 $D n_M$ 被替换为分段仿射边方向场的分段梯度。我们将无假设渐近下界（适用于任何用三角复形对曲面进行的一致逼近）与恢复序列相结合，该恢复序列由极限序列的任意正则三角剖分和边方向的几乎最优选取构成。