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Constructions and conjectures in crystalline nondifferential geometry. (English) Zbl 0725.53011
Differential geometry. A symposium in honour of Manfredo do Carmo, Proc. Int. Conf., Rio de Janeiro/Bras. 1988, Pitman Monogr. Surv. Pure Appl. Math. 52, 321-336 (1991).
[For the entire collection see Zbl 0718.00010.]
This survey discusses some crystalline problems involving surface energy minimization analogous to corresponding surface area minimization problems. The surface energy density F per unit surface area varies with the oriented normal direction \(\nu\) of the crystal’s bounding surface. F determines the Wulff shape W of the crystal material: the intersection of the half spaces \(\{x:\;x\cdot \nu \leq F(\nu),\text{ for all } \nu \}.\) In the isotropic case F is constant, W is a ball of radius F, and the total surface energy is F times the surface area.
Here are several typical results and conjectures from the paper. Consider a crystal touching the bounding plane of a half-space in the absence of gravity. For constant F, the case of a fluid drop, the equilibrium figure is part of a ball of radius F. For a crystal, this ball is replaced by W. In the presence of gravity, a gravity-induced facet may occur which W does not exhibit. Conjecture: In the presence of gravity the equilibrium facet is convex. By analogy with Plateau problems, when W is polyhedral and a closed polygonal curve C is given, subject to some restrictions, there is a construction producing a surface which minimizes total surface energy among all polyhedral surfaces of the same topological type which span C. The paper closes with a discussion of crystalline analogues of motion by mean curvature as discussed, for example, by M. Gage and R. Hamilton [J. Differ. Geom. 23, 69-96 (1986; Zbl 0621.53001)].

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C80 Applications of global differential geometry to the sciences
51P05 Classical or axiomatic geometry and physics (should also be assigned at least one other classification number from Sections 70-XX–86-XX)