The affine Plateau problem

Trudinger, Neil; Wang, Xu-Jia

doi:10.1090/S0894-0347-05-00475-3

The affine Plateau problem

Authors: Neil S. Trudinger and Xu-Jia Wang
Journal: J. Amer. Math. Soc. 18 (2005), 253-289
MSC (2000): Primary 35J40; Secondary 53A15
DOI: https://doi.org/10.1090/S0894-0347-05-00475-3
Published electronically: January 3, 2005
MathSciNet review: 2137978
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Abstract: In this paper, we study a second order variational problem for locally convex hypersurfaces, which is the affine invariant analogue of the classical Plateau problem for minimal surfaces. We prove existence, regularity and uniqueness results for hypersurfaces maximizing affine area under appropriate boundary conditions.

1 W. Blaschke, Vorlesungen über Differential geometrie, Berlin, 1923.

L. A. Caffarelli, A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity, Ann. of Math. (2) 131 (1990), no. 1, 129–134. MR 1038359, DOI https://doi.org/10.2307/1971509
Luis A. Caffarelli, Interior $W^{2,p}$ estimates for solutions of the Monge-Ampère equation, Ann. of Math. (2) 131 (1990), no. 1, 135–150. MR 1038360, DOI https://doi.org/10.2307/1971510
Luis A. Caffarelli and Cristian E. Gutiérrez, Properties of the solutions of the linearized Monge-Ampère equation, Amer. J. Math. 119 (1997), no. 2, 423–465. MR 1439555
L. Caffarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equation, Comm. Pure Appl. Math. 37 (1984), no. 3, 369–402. MR 739925, DOI https://doi.org/10.1002/cpa.3160370306
Eugenio Calabi, Hypersurfaces with maximal affinely invariant area, Amer. J. Math. 104 (1982), no. 1, 91–126. MR 648482, DOI https://doi.org/10.2307/2374069
Eugenio Calabi, Affine differential geometry and holomorphic curves, Complex geometry and analysis (Pisa, 1988) Lecture Notes in Math., vol. 1422, Springer, Berlin, 1990, pp. 15–21. MR 1055839, DOI https://doi.org/10.1007/BFb0089401
Shiu Yuen Cheng and Shing-Tung Yau, Complete affine hypersurfaces. I. The completeness of affine metrics, Comm. Pure Appl. Math. 39 (1986), no. 6, 839–866. MR 859275, DOI https://doi.org/10.1002/cpa.3160390606
Shiing Shen Chern, Affine minimal hypersurfaces, Minimal submanifolds and geodesics (Proc. Japan-United States Sem., Tokyo, 1977) North-Holland, Amsterdam-New York, 1979, pp. 17–30. MR 574250
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
Bo Guan and Joel Spruck, Boundary-value problems on $S^n$ for surfaces of constant Gauss curvature, Ann. of Math. (2) 138 (1993), no. 3, 601–624. MR 1247995, DOI https://doi.org/10.2307/2946558
N. M. Ivočkina, A priori estimate of $|u|_{C_2(\overline \Omega )}$ of convex solutions of the Dirichlet problem for the Monge-Ampère equation, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 96 (1980), 69–79, 306 (Russian). Boundary value problems of mathematical physics and related questions in the theory of functions, 12. MR 579472
N. V. Krylov, Nonlinear elliptic and parabolic equations of the second order, Mathematics and its Applications (Soviet Series), vol. 7, D. Reidel Publishing Co., Dordrecht, 1987. Translated from the Russian by P. L. Buzytsky [P. L. Buzytskiĭ]. MR 901759
Kurt Leichtweiß, Affine geometry of convex bodies, Johann Ambrosius Barth Verlag, Heidelberg, 1998. MR 1630116
Erwin Lutwak, Extended affine surface area, Adv. Math. 85 (1991), no. 1, 39–68. MR 1087796, DOI https://doi.org/10.1016/0001-8708%2891%2990049-D
Katsumi Nomizu and Takeshi Sasaki, Affine differential geometry, Cambridge Tracts in Mathematics, vol. 111, Cambridge University Press, Cambridge, 1994. Geometry of affine immersions. MR 1311248
Aleksey Vasil′yevich Pogorelov, The Minkowski multidimensional problem, V. H. Winston & Sons, Washington, D.C.; Halsted Press [John Wiley & Sons], New York-Toronto-London, 1978. Translated from the Russian by Vladimir Oliker; Introduction by Louis Nirenberg; Scripta Series in Mathematics. MR 0478079
M. V. Safonov, Classical solution of second-order nonlinear elliptic equations, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 6, 1272–1287, 1328 (Russian); English transl., Math. USSR-Izv. 33 (1989), no. 3, 597–612. MR 984219, DOI https://doi.org/10.1070/IM1989v033n03ABEH000858

19 W.M. Sheng, N.S. Trudinger, and X.-J. Wang, Enclosed convex hypersurfaces with maximal affine area, preprint.

Udo Simon, Affine differential geometry, Handbook of differential geometry, Vol. I, North-Holland, Amsterdam, 2000, pp. 905–961. MR 1736860, DOI https://doi.org/10.1016/S1874-5741%2800%2980012-6
Neil S. Trudinger and Xu-Jia Wang, The Bernstein problem for affine maximal hypersurfaces, Invent. Math. 140 (2000), no. 2, 399–422. MR 1757001, DOI https://doi.org/10.1007/s002220000059
Neil S. Trudinger and Xu-Jia Wang, On locally convex hypersurfaces with boundary, J. Reine Angew. Math. 551 (2002), 11–32. MR 1932171, DOI https://doi.org/10.1515/crll.2002.078

23 N.S. Trudinger and X.-J. Wang, On boundary regularity for the Monge-Ampère and affine maximal surface equations, preprint.

Xu-Jia Wang, Affine maximal hypersurfaces, Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 221–231. MR 1957534
William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. MR 1014685