Abstract
In our previous paper [4] we have investigated level surfaces of a non-degenerate function ϕ in a real affine space An+1 by using the gradient vector field\(\tilde E\). We gave characterizations of ϕ by means of the shape operatorS, the transversal connection τ, and studied the difference between\(\tilde E\) and the affine normal. In particular we showed that a graph defined by a non-degenerate function satisfiesS=0 and τ=0. In this paper we consider harmonic gradient mappings of level surfaces and apply these results to a certain problem which is similar to the affine Bernstein problem conjectured by S. S. Chern [3].
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Shima, H. Harmonicity of gradient mappings of level surfaces in a real affine space. Geom Dedicata 56, 177–184 (1995). https://doi.org/10.1007/BF01267641
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DOI: https://doi.org/10.1007/BF01267641