Given an exterior domain Ω \Omega with C 2 , α C^{2,\alpha } boundary in R n \mathbb {R}^{n} , n ≥ 3 n\geq 3 , we obtain a 1 1 -parameter family u γ ∈ C ∞ ( Ω ) u_{\gamma }\in C^{\infty }\left ( \Omega \right ) , | γ | ≤ π / 2 \left \vert \gamma \right \vert \leq \pi /2 , of solutions of the minimal surface equation such that, if | γ | > π / 2 \left \vert \gamma \right \vert >\pi /2 , u γ ∈ C ∞ ( Ω ) ∩ C 2 , α ( Ω ¯ ) u_{\gamma }\in C^{\infty }\left ( \Omega \right ) \cap C^{2,\alpha }\left ( \overline {\Omega }\right ) , u γ | ∂ Ω = 0 u_{\gamma }|_{\partial \Omega }=0 with max ∂ Ω ‖ ∇ u γ ‖ = tan ⁡ γ \max _{\partial \Omega }\left \Vert \nabla u_{\gamma }\right \Vert =\tan \gamma and, if | γ | = π / 2 \left \vert \gamma \right \vert =\pi /2 , the graph of u γ u_{\gamma } is contained in a C 1 , 1 C^{1,1} manifold M γ ⊂ Ω ¯ × R M_{\gamma }\subset \overline {\Omega }\times \mathbb {R} with ∂ M γ = ∂ Ω \partial M_{\gamma }=\partial \Omega . Each of these functions is bounded and asymptotic to a constant \[ c γ = lim ‖ x ‖ → ∞ u γ ( x ) . c_{\gamma }=\lim _{\left \Vert x\right \Vert \rightarrow \infty }u_{\gamma }\left ( x\right ) . \] The mappings γ → u γ ( x ) \gamma \rightarrow u_{\gamma }\left ( x\right ) (for fixed x ∈ Ω x\in \Omega ) and γ → c γ \gamma \rightarrow c_{\gamma } are strictly increasing and bounded. The graphs of these functions foliate the open subset of R n + 1 \mathbb {R}^{n+1} \[ { ( x , z ) ∈ Ω × R ,  − u π / 2 ( x ) > z > u π / 2 ( x ) } . \left \{ \left ( x,z\right ) \in \Omega \times \mathbb {R}\text {, }-u_{\pi /2}\left ( x\right ) >z>u_{\pi /2}\left ( x\right ) \right \} . \] Moreover, if R n ∖ Ω \mathbb {R}^{n}\backslash \Omega satisfies the interior sphere condition of maximal radius ρ \rho and if ∂ Ω \partial \Omega is contained in a ball of minimal radius ϱ \varrho , then \[ [ 0 , σ n ρ ] ⊂ [ 0 , c π / 2 ] ⊂ [ 0 , σ n ϱ ] , \left [ 0,\sigma _{n}\rho \right ] \subset \left [ 0,c_{\pi /2}\right ] \subset \left [ 0,\sigma _{n}\varrho \right ] , \] where \[ σ n = ∫ 1 ∞ d t t 2 ( n − 1 ) − 1 . \sigma _{n}=\int _{1}^{\infty }\frac {dt}{\sqrt {t^{2\left ( n-1\right ) }-1}}. \] One of the above inclusions is an equality if and only if ρ = ϱ \rho =\varrho , Ω \Omega is the exterior of a ball of radius ρ \rho and the solutions are radial. These foliations were studied by E. Kuwert in [Ann. Inst. H. Poincaré Anal. Non Linéaire 10 (1993), pp. 445–451] and our result answers a natural question about the existence of such foliations which was not touched by Kuwert.

It is proven results about existence and nonexistence of unit normal sections of submanifolds of the Euclidean space and sphere, which associated Gauss maps, are harmonic. Some applications to constant mean curvature hypersurfaces of the sphere and to isoparametric submanifolds are obtained too. K E Y W O R D Sharmonic Gauss maps, isoparametric submanifolds, minimal surfaces, surfaces with parallel mean curvature1 Theorem 1.1. Let 𝑀 be a surface of ℝ 4 with parallel mean curvature vector field such that the second fundamental form of 𝑀 spans the normal space of 𝑀 in ℝ 4 at each point. Then, any unit normal section of 𝑀 in ℝ 4 can be written as 𝜂 = 𝑎𝜈 + 𝑏𝜇,

No abstract