Minimal surface systems, maximal surface systems and special Lagrangian equations
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- by Hojoo Lee PDF
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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
- MR Author ID: 692348
- Email: ultrametric@gmail.com, autumn@kias.re.kr
- 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.
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 3775-3797
- MSC (2010): Primary 49Q05, 35J47, 35B08, 53D12
- DOI: https://doi.org/10.1090/S0002-9947-2012-05786-2
- MathSciNet review: 3042603