Harmonic minimal graphs in product spaces

Abstract It is known that a harmonic minimal graph in 3-dimensional Euclidean space is locally either a plane or a helicoid. We prove that any harmonic minimal graph in $${{\mathbb {R}}}^{n+1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> is locally foliated by $$(n-1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -dimensional minimal graphs. Moreover, it is proved that a harmonic minimal graph in $$M^{2}\times {{\mathbb {R}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>×</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> is locally either a horizontal level surface or a helicoid, where $$M^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> denotes either 2-dimensional hyperbolic space $${\mathbb {H}}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> or the 2-dimensional unit sphere $${\mathbb {S}}^{2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> .