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Convergence of a three-dimensional crystalline motion to Gauss curvature flow. (English) Zbl 1089.53046
Authors’ summary: We introduce a three-dimensional crystalline motion whose Wulff shape is a convex polyhedron $$(\widetilde W^k)$$. We prove that this crystalline motion converges to the motion by Gauss curvature in $$\mathbb{R}^3$$ under the assumptions that the polyhedra $$\widetilde W^k$$ converge to the unit ball $$B^3$$ and are symmetric with respect to the origin.
K. Ishii and H. M. Soner showed the convergence of the two-dimensional crystalline motion to the curve shortening flow by a kind of perturbed test function methods. We employ their method to prove our result under aid from the theory of the Minkowski problem.

##### MSC:
 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) 53C80 Applications of global differential geometry to the sciences 82D25 Statistical mechanical studies of crystals
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