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Physics and geometry based variational techniques for surface construction have been shown to be advanced methods for designing high quality surfaces in the fields of CAD and CAGD. In this paper, we derive an Euler-Lagrange equation from a geometric invariant curvature integral functional-the integral about the mean curvature gradient. Using this Euler-Lagrange equation, we construct a sixth-order geometric flow, which is solved numerically by a divided-difference-like method. We apply our equation to solving several surface modeling problems, including surface blending, N-sided hole filling and point interpolating, with G^2 continuity. The illustrative examples provided show that this sixth-order flow yields high quality surfaces.