Abstract
Instead on the slope function, we establish a condition on the singular values of the differential of a vector-valued function \(f:\mathbb {R}^{n} \to \mathbb {R}^{m}\) which ensures that such f satisfying the minimal surface equations is affine linear. This is a Bernstein type theorem for entire minimal graphs of arbitrary dimension and codimension, improving the results in Jost and **n (Calc. Var. PDE 9, 277–296, 1999) and Wang (Trans. Amer. Math. Soc. 355, 265–271, 2003). The proof depends on the superharmoncity of an auxiliary function and on conclusions from geometric measure theory.
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Acknowledgments
This paper was inspired by a private discussion with Prof. Y. L. **n [26], who conjectured that every entire minimal graph has to be planar, whenever ∥Λ2df∥≤ 2 and Δf ≤ C with a positive constant C. (The Bernstein type theorem in [15] can be directly deduced from this conjecture.) The second named author wishes to thank Prof. J. Jost and Prof. Y. L. **n, for their encouragement and support. He is also partially supported by NSFC (No. 11471078, 11622103).
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Dedicated to Prof. J. Jost on his 65th birthday.
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**g, L., Yang, L. A Bernstein-Type Theorem for Minimal Graphs of Higher Codimension via Singular Values. Vietnam J. Math. 49, 481–492 (2021). https://doi.org/10.1007/s10013-021-00473-z
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DOI: https://doi.org/10.1007/s10013-021-00473-z