A Bernstein type theorem for minimal volume preserving maps (Q2781272)
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scientific article; zbMATH DE number 1721018
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | A Bernstein type theorem for minimal volume preserving maps |
scientific article; zbMATH DE number 1721018 |
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A Bernstein type theorem for minimal volume preserving maps (English)
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19 March 2002
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minimal maps
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volume preserving
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lagrangian submanifolds
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JFM 48.1401.01
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0.92368734
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0.90394133
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0.90293324
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0.89767516
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0.8855653
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0.88170135
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0.8798502
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A map between two Riemannian manifolds \(M_1,M_2\) is called minimal if its graph is a minimal submanifold of \(M_1\times M_2\). Bernstein's theorem [\textit{S. Bernstein}, Char'kov, Comm. Soc. Math. 15, 38-45 (1917; JFM 48.1401.01)] (stating that any minimal map from \(\mathbb{R}^2\) to \(\mathbb{R}\) must be linear), was generalized by several authors. One of them is the present author, who derives here that any minimal volume preserving map from \(\mathbb{R}^2\) into \(\mathbb{R}^2\) is a linear diffeomorphism, by using that any minimal diffeomorphism of \(\mathbb{R}^2\) is linear.
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