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)].

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)].

Reviewer: W.J.Firey (Corvallis)

##### MSC:

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) |