Summary
In the past fifteen years or so, convex geometry and the theory of calibrations have provided a deeper understanding of the behavior and singular structure ofm-dimensional area-minimizing surfaces inR n. Calibrations correspond to faces of the GrassmannianG(m,R n) of orientedm-planes inR n, viewed as a compact submanifold of the exterior algebra Λm R n. Large faces typically provide many examples of area-minimizing surfaces. This paper studies the sizes of such faces. It also considers integrands Φ more general than area. One result implies that form-dimensional surfaces inR n, with 2 ⩽m ⩽n − 2, for any integrand Φ, there are Φ-minimizing surfaces with interior singularities.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Dadok, J., Harvey, R. andMorgan, F.,Calibrations on R 8. Trans. Amer. Math. Soc.307 (1988), 1–40.
De Rham, G.,On the area of complex manifolds. Notes for the Seminar on Several Complex Variables. Inst. for Adv. Study, Princeton, 1957–58.
Federer, H.,Geometric measure theory. Springer-Verlag, New York, 1969.
Federer, H.,Some theorems on integral currents. Trans. Amer. Math. Soc.117 (1965), 43–67.
Harvey, R. andLawson, H. B., Jr.,Calibrated geometries. Acta Math.148 (1982), 47–157.
Harvey, R. andMorgan, F.,The face of the Grassmannian of three-planes in R 7. Invent. Math.83 (1986), 191–228.
Lansberg, J. M.,Minimal submanifolds defined by first order systems of PDE. Ph.D. dissertation, Duke U., 1990.
Morgan, F.,Almost every curve in R 3 bounds a unique area minimizing surface. Invent. Math.45 (1978), 253–297.
Morgan, F.,Area-minimizing surfaces, faces of Grassmannians, and calibrations. Amer. Math. Monthly95 (1988), 813–822.
Morgan, F.,Calibrations and new singularities in area-minimizing surfaces: a survey. InVariational methods (Proc. Conf. Paris, June, 1988). [Prog. Nonlinear Diff. Eqns. Applns. Vol. 4]. Birkhāuser, Boston, 1990, pp. 329–342.
Morgan, F.,Examples of unoriented area-minimizing surfaces. Trans. Amer. Math. Soc.283 (1984), 225–237.
Morgan, F.,The exterior algebra and area minimization. Linear Algebra Appl.66 (1985), 1–28.
Morgan, F.,Geometric measure theory: a beginner's guide. Academic Press, Boston, 1988.
Morgan, F.,Measures on spaces of surfaces. Arch. Rational Mech. Anal.78 (1982), 335–359.
Morgan, F.,Minimal surfaces, crystals, shortest networks, and undergraduate research. Math. Intelligencer, to appear.
Morgan, F.,On the singular structure of three-dimensional area-minimizing surfaces in R n. Trans. Amer. Math Soc.275 (1983) 137–143.
Morgan, F.,Riemannian geometry: a beginner's guide. Jones and Bartlett, Boston, to appear.
Taylor, J.,Crystalline variational problems. Bull. Amer. Math. Soc.84 (1987), 568–588.
White, B.,The space of m-dimensional surfaces that are stationary for a parametric elliptic functional. Indiana Univ. Math. J.36 (1987), 567–602.
Wirtinger, W.,Eine Determinantenidentitat und ihre Anwendung auf analytische Gebilde und Hermitesche Massbestimmung. Monatsh. Math. Physik44 (1936), 343–365.