Calibrations associated to Monge-Ampère equations

Warren, Micah

doi:10.1090/S0002-9947-10-05109-3

Calibrations associated to Monge-Ampère equations
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by Micah Warren PDF

Trans. Amer. Math. Soc. 362 (2010), 3947-3962 Request permission

Abstract:

We show the volume maximizing property of the special Lagrangian submanifolds of a pseudo-Euclidean space. These special Lagrangian submanifolds arise locally as gradient graphs of solutions to Monge-Ampère equations.

References

Eugenio Calabi, Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens, Michigan Math. J. 5 (1958), 105–126. MR 106487
Vicente Cortés, Christoph Mayer, Thomas Mohaupt, and Frank Saueressig, Special geometry of Euclidean supersymmetry. I. Vector multiplets, J. High Energy Phys. 3 (2004), 028, 73. MR 2061551, DOI 10.1088/1126-6708/2004/03/028
Harley Flanders, On certain functions with positive definite Hessian, Ann. of Math. (2) 71 (1960), 153–156. MR 145025, DOI 10.2307/1969882
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
Reese Harvey and H. Blaine Lawson Jr., Calibrated geometries, Acta Math. 148 (1982), 47–157. MR 666108, DOI 10.1007/BF02392726
Nigel J. Hitchin, The moduli space of special Lagrangian submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 3-4, 503–515 (1998). Dedicated to Ennio De Giorgi. MR 1655530
Konrad Jörgens, Über die Lösungen der Differentialgleichung $rt-s^2=1$, Math. Ann. 127 (1954), 130–134 (German). MR 62326, DOI 10.1007/BF01361114
J. Jost and Y. L. Xin, Bernstein type theorems for higher codimension, Calc. Var. Partial Differential Equations 9 (1999), no. 4, 277–296. MR 1731468, DOI 10.1007/s005260050142
Young-Heon Kim and Robert J. McCann, Continuity, curvature, and the general covariance of optimal transportation, Journal of the European Mathematical Society (To Appear).
Young-Heon Kim, Robert J. McCann, and Micah Warren, Pseudo-Riemannian geometry calibrates optimal transportation, Preprint.
Jack Mealy, Volume maximization in semi-Riemannian manifolds, Indiana Univ. Math. J. 40 (1991), no. 3, 793–814. MR 1129330, DOI 10.1512/iumj.1991.40.40036
Robert C. McLean, Deformations of calibrated submanifolds, Comm. Anal. Geom. 6 (1998), no. 4, 705–747. MR 1664890, DOI 10.4310/CAG.1998.v6.n4.a4
A. V. Pogorelov, On the improper convex affine hyperspheres, Geometriae Dedicata 1 (1972), no. 1, 33–46. MR 319126, DOI 10.1007/BF00147379
Yu Yuan, A Bernstein problem for special Lagrangian equations, Invent. Math. 150 (2002), no. 1, 117–125. MR 1930884, DOI 10.1007/s00222-002-0232-0
—, Global solutions to special Lagrangian equations, Proc. Amer. Math. Soc. 134 (2006), no. 5, 1355–1358.