The Bernstein problem for complete Lagrangian stationary surfaces
Author: Chikako Mese
Journal: Proc. Amer. Math. Soc. 129 (2001), 573-580
MSC (1991): Primary 58E12; Secondary 53C15
Published electronically: July 27, 2000
MathSciNet review: 1707155
Full-text PDF Free Access
In this paper, we investigate the global geometric behavior of lagrangian stationary surfaces which are lagrangian surfaces whose area is critical with respect to lagrangian variations. We find that if a complete oriented immersed lagrangian surface has quadratic area growth, one end and finite topological type, then it is minimal and hence holomorphic. The key to the proof is the mean curvature estimate of Schoen and Wolfson combined with the observation that a complete immersed surface of quadratic area growth, finite topology and mean curvature has finite total absolute curvature.
- [BdC] J. L. Barbosa and M. do Carmo, On the size of a stable minimal surface in 𝑅³, Amer. J. Math. 98 (1976), no. 2, 515–528. MR 0413172, https://doi.org/10.2307/2373899
- [B] Robert L. Bryant, Minimal Lagrangian submanifolds of Kähler-Einstein manifolds, Differential geometry and differential equations (Shanghai, 1985) Lecture Notes in Math., vol. 1255, Springer, Berlin, 1987, pp. 1–12. MR 895393, https://doi.org/10.1007/BFb0077676
- [dCP] J. L. Barbosa and M. do Carmo, On the size of a stable minimal surface in 𝑅³, Amer. J. Math. 98 (1976), no. 2, 515–528. MR 0413172, https://doi.org/10.2307/2373899
- [Ch] Qing Chen, On the total curvature and area growth of minimal surfaces in 𝐑ⁿ, Manuscripta Math. 92 (1997), no. 2, 135–142. MR 1428644, https://doi.org/10.1007/BF02678185
- [C] Shiing-shen Chern, Minimal surfaces in an Euclidean space of 𝑁 dimensions, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 187–198. MR 0180926
- [CO] Shiing-shen Chern and Robert Osserman, Complete minimal surfaces in euclidean 𝑛-space, J. Analyse Math. 19 (1967), 15–34. MR 0226514, https://doi.org/10.1007/BF02788707
- [FS] Doris Fischer-Colbrie and Richard Schoen, The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33 (1980), no. 2, 199–211. MR 562550, https://doi.org/10.1002/cpa.3160330206
- [HL] Reese Harvey and H. Blaine Lawson Jr., Calibrated geometries, Acta Math. 148 (1982), 47–157. MR 666108, https://doi.org/10.1007/BF02392726
- [H] A. Huber, On Subharmonic Functions and Differential Geometry in the Large, Comm. Math. Helv., 32(1966), 105-136.
- [Li] Peter Li, Complete surfaces of at most quadratic area growth, Comment. Math. Helv. 72 (1997), no. 1, 67–71. MR 1456316, https://doi.org/10.1007/PL00000367
- [M] Mario J. Micallef, Stable minimal surfaces in Euclidean space, J. Differential Geom. 19 (1984), no. 1, 57–84. MR 739782
- [Oh] Yong-Geun Oh, Second variation and stabilities of minimal Lagrangian submanifolds in Kähler manifolds, Invent. Math. 101 (1990), no. 2, 501–519. MR 1062973, https://doi.org/10.1007/BF01231513
- [O1] Robert Osserman, Proof of a conjecture of Nirenberg, Comm. Pure Appl. Math. 12 (1959), 229–232. MR 0105700, https://doi.org/10.1002/cpa.3160120203
- [O2] Robert Osserman, On complete minimal surfaces, Arch. Rational Mech. Anal. 13 (1963), 392–404. MR 0151907, https://doi.org/10.1007/BF01262706
- [O3] Robert Osserman, Global properties of minimal surfaces in 𝐸³ and 𝐸ⁿ, Ann. of Math. (2) 80 (1964), 340–364. MR 0179701, https://doi.org/10.2307/1970396
- [SSY] R. Schoen, L. Simon, and S. T. Yau, Curvature estimates for minimal hypersurfaces, Acta Math. 134 (1975), no. 3-4, 275–288. MR 0423263, https://doi.org/10.1007/BF02392104
- [SW] Richard Schoen and Jon Wolfson, Minimizing volume among Lagrangian submanifolds, Differential equations: La Pietra 1996 (Florence), Proc. Sympos. Pure Math., vol. 65, Amer. Math. Soc., Providence, RI, 1999, pp. 181–199. MR 1662755, https://doi.org/10.1090/pspum/065/1662755
- [W] J. Wolfson, Minimal Lagrangian Diffeomeorphisms and the Monge-Ampére Equation, to appear J. Diff. Geo.
- [X] F. Xavier, The Gauss Map of a Complete Non-flat Minimal Surfaces Cannot Omit 7 Points on the Sphere, Ann. of Math. 113(1981) 211-214.
- J.L. Barbosa and M. doCarmo, On the Size of a Stable Minimal Surface in , Amer. J. Math 98(1976) 515-528.MR 54:1292
- R. Bryant, Minimal Lagrangian Submanifolds of Kähler-Einstein Manifolds, Lecture Notes in Math. 1255, 1-12, Springer-Verlag, New York, 1987.MR 88j:53061
- M. doCarmo and C.K. Peng, Stable Minimal Surfaces in are planes, Bull. Amer. Math., 98 (1976) 515-528.MR 54:1292
- Q. Chen, On the Total Curvature and Area Growth of Minimal Surfaces in , Manu. Math., 92(1997) 135-142.MR 98c:49073
- S.S. Chern, Minimal Surfaces in an Euclidean Space of N Dimensions, Differential and Combinational Topology, A symposium in honor of Marston Morse, Princeton University Press, Princeton, 1965, 187-198.MR 31:5156
- S.S. Chern and R. Osserman, Complete Minimal Surfaces in Euclidean n-Space, J. Analyse Math. 19(1967) 15-34. MR 37:2103
- D. Fischer-Colbrie and R. Schoen, The Structure of Complete Stable Minimal Surfaces in 3-manifolds of Non-negative Scalar Curvature, Comm. Pure Appl. Math 33(1980) 199-211.MR 81i:53044
- R. Harvey and H. B. Lawson, Calibrated Geoemtries, Acta Math., 148(1982) 47-157. MR 85i:53058
- A. Huber, On Subharmonic Functions and Differential Geometry in the Large, Comm. Math. Helv., 32(1966), 105-136.
- P. Li, Complete Surfaces of at Most Quadratic Area Growth, Comm. Math. Helv., 72(1997) 67-71. MR 98h:53057
- M. Micallef, Stable Minimal Surfaces in Euclidean Space, J. Diff. Geo, (1994), 57-84. MR 85e:53009
- Y.G.Oh, Second Variation and Stabilities of Minimal Lagrangian Submanifolds in Kähler Manifolds, Invent. Math., 101(1990), 501-519.MR 91f:58022
- R. Osserman, Proof of a Conjecture of Nirenberg, Comm. Pure Appl. Math. 12(1959) 229-232. MR 21:4436
- R. Osserman, On Complete Minimal Surfaces, Arch. Rational Mech. Anal., 13(1963) 392-404. MR 27:1888
- R. Osserman, Global Properties of Minimal Surfaces in and . Ann. of Math. 2, 80(1964) 340-364. MR 31:3946
- R. Schoen, L. Simon and S.T.Yau, Curvature Estimates for Minimal Hypersurfaces, Acta Math., 134(1975) 275-288.MR 54:11243
- R. Schoen and J Wolfson, Minimizing Volume Among Lagrangian Submanifolds. preprint. MR 99k:53130
- J. Wolfson, Minimal Lagrangian Diffeomeorphisms and the Monge-Ampére Equation, to appear J. Diff. Geo.
- F. Xavier, The Gauss Map of a Complete Non-flat Minimal Surfaces Cannot Omit 7 Points on the Sphere, Ann. of Math. 113(1981) 211-214.
Affiliation: Department of Mathematics DRB155, University of Southern California, 1042 West 36th Place, Los Angeles, California 90089
Address at time of publication: Box 5657, Department of Mathematics, Connecticut College, 270 Mohegan Ave., New London, Connecticut 06320
Received by editor(s): April 12, 1999
Published electronically: July 27, 2000
Additional Notes: The author would like to thank Professor Richard Schoen for introducing her to this problem and Professor Paul Yang for his interest in this work. Additionally, she thanks Professor Francis Bonahon for many useful conversations.
Communicated by: Bennett Chow
Article copyright: © Copyright 2000 American Mathematical Society