Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



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
Published electronically: January 3, 2005
MathSciNet review: 2137978
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. W. Blaschke, Vorlesungen über Differential geometrie, Berlin, 1923.
  • 2. 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, 10.2307/1971509
  • 3. 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, 10.2307/1971510
  • 4. 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
  • 5. 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, 10.1002/cpa.3160370306
  • 6. Eugenio Calabi, Hypersurfaces with maximal affinely invariant area, Amer. J. Math. 104 (1982), no. 1, 91–126. MR 648482, 10.2307/2374069
  • 7. 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, 10.1007/BFb0089401
  • 8. 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, 10.1002/cpa.3160390606
  • 9. 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
  • 10. 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
  • 11. Bo Guan and Joel Spruck, Boundary-value problems on 𝑆ⁿ for surfaces of constant Gauss curvature, Ann. of Math. (2) 138 (1993), no. 3, 601–624. MR 1247995, 10.2307/2946558
  • 12. N. M. Ivočkina, A priori estimate of \vert𝑢\vert_{𝐶₂(\overlineΩ)} 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
  • 13. 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
  • 14. Kurt Leichtweiß, Affine geometry of convex bodies, Johann Ambrosius Barth Verlag, Heidelberg, 1998. MR 1630116
  • 15. Erwin Lutwak, Extended affine surface area, Adv. Math. 85 (1991), no. 1, 39–68. MR 1087796, 10.1016/0001-8708(91)90049-D
  • 16. 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
  • 17. 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
  • 18. 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
  • 19. W.M. Sheng, N.S. Trudinger, and X.-J. Wang, Enclosed convex hypersurfaces with maximal affine area, preprint.
  • 20. Udo Simon, Affine differential geometry, Handbook of differential geometry, Vol. I, North-Holland, Amsterdam, 2000, pp. 905–961. MR 1736860, 10.1016/S1874-5741(00)80012-6
  • 21. Neil S. Trudinger and Xu-Jia Wang, The Bernstein problem for affine maximal hypersurfaces, Invent. Math. 140 (2000), no. 2, 399–422. MR 1757001, 10.1007/s002220000059
  • 22. Neil S. Trudinger and Xu-Jia Wang, On locally convex hypersurfaces with boundary, J. Reine Angew. Math. 551 (2002), 11–32. MR 1932171, 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.
  • 24. 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
  • 25. 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

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 35J40, 53A15

Retrieve articles in all journals with MSC (2000): 35J40, 53A15

Additional Information

Neil S. Trudinger
Affiliation: Centre for Mathematics and Its Applications, The Australian National University, Canberra, ACT 0200, Australia

Xu-Jia Wang
Affiliation: Centre for Mathematics and Its Applications, Australian National University, Canberra, ACT 0200, Australia

Keywords: Affine Plateau problem, affine maximal hypersurface, affine area functional, affine maximal surface equation, variational problem, second boundary value problem, a priori estimates, strict convexity, interior regularity, Bernstein Theorem, Monge-Amp\`{e}re measure, curvature measure, Gauss mapping, locally convex hypersurface, generalized Legendre transform
Received by editor(s): September 3, 2003
Published electronically: January 3, 2005
Additional Notes: This research was supported by the Australian Research Council
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.