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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Minimal surface systems, maximal surface systems and special Lagrangian equations


Author: Hojoo Lee
Journal: Trans. Amer. Math. Soc. 365 (2013), 3775-3797
MSC (2010): Primary 49Q05, 35J47, 35B08, 53D12
Published electronically: November 7, 2012
MathSciNet review: 3042603
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Abstract: We extend Calabi's correspondence between minimal graphs in Euclidean space $ {\mathbb{R}}^{3}$ and maximal graphs in Lorentz-Minkowski space $ {\mathbb{L}}^{3}$. We establish the twin correspondence between $ 2$-dimensional minimal graphs in Euclidean space $ {\mathbb{R}}^{n+2}$ carrying a positive area-angle function and $ 2$-dimension-
al maximal graphs in pseudo-Euclidean space $ {\mathbb{R}}^{n+2}_{n}$ carrying the same positive area-angle function.

We generalize Osserman's Lemma on degenerate Gauss maps of entire $ 2$-dimensional minimal graphs in $ {\mathbb{R}}^{n+2}$ and offer several Bernstein-Calabi type theorems. A simultaneous application of the Harvey-Lawson Theorem on special Lagrangian equations and our extended Osserman's Lemma yield a geometric proof of Jörgens' Theorem on the $ 2$-variable unimodular Hessian equation.

We introduce the correspondence from $ 2$-dimensional minimal graphs in $ {\mathbb{R}}^{n+2}$ to special Lagrangian graphs in $ {\mathbb{C}}^{2}$, which induces an explicit correspondence from $ 2$-variable symplectic Monge-Ampère equations to the $ 2$-variable unimodular Hessian equation.


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Additional Information

Hojoo Lee
Affiliation: Department of Geometry and Topology, University of Granada, Granada, Spain
Address at time of publication: School of Mathematics, Korea Institute for Advanced Study, 207-43 Cheongryangri 2-dong, Dongdaemun-gu, Seoul 130-722, Korea
Email: ultrametric@gmail.com, autumn@kias.re.kr

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05786-2
PII: S 0002-9947(2012)05786-2
Keywords: Minimal submanifold, special Lagrangian equation, entire solution
Received by editor(s): December 27, 2010
Received by editor(s) in revised form: July 20, 2011, and November 11, 2011
Published electronically: November 7, 2012
Additional Notes: This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (Ministry of Education, Science and Technology) [NRF-2011-357-C00007] and in part by 2010 Korea-France STAR Program.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.