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Questions and answers about area-minimizing surfaces and geometric measure theory. (English) Zbl 0812.49032
Greene, Robert (ed.) et al., Differential geometry. Part 1: Partial differential equations on manifolds. Proceedings of a summer research institute, held at the University of California, Los Angeles, CA, USA, July 8-28, 1990. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 54, Part 1, 29-53 (1993).
The author offers “informal personal answers” to 40 questions “which are sometimes asked about area minimization or geometric measure theory”. Examples are: What is an area-minimizing surface? What difference does the definition of area make? What kinds of area- minimizing surfaces can be computed? Have minimal surface computations been useful outside mathematics? Why are least area problems hard theoretically? What are the main successes of modern geometric measure theory? What are currents, why are they called that, and where did they come from? What are some of the main theorems about integral currents? What is the uniqueness of tangent cones problem? What are multifunctions and what are they good for? What are that chains mod ν? What are crystalline minimal surfaces? What are calibrations? There are 80 references.
MSC:
49Q20Variational problems in a geometric measure-theoretic setting
28A75Length, area, volume, other geometric measure theory
32C30Integration on analytic sets and spaces, currents
52A27Approximation by convex sets
74A55Theories of friction (tribology)
74M15Contact (solid mechanics)
53A10Minimal surfaces, surfaces with prescribed mean curvature
65D18Computer graphics, image analysis, and computational geometry