Proper holomorphic embeddings with small limit sets

Let X X be a Stein manifold of dimension n ≥<!-- ≥ --> 1 n≥ 1 . Given a continuous positive increasing function h h on R + = [ 0 , ∞<!-- ∞ --> ) R_+=[0,∞ ) with lim t →<!-- → --> ∞<!-- ∞ --> h ( t ) = ∞<!-- ∞ --> lim t→ ∞ h(t)=∞ , we construct a proper holomorphic embedding f = ( z , w ) : X ↪<!-- ↪ --> C n + 1 ×<!-- × --> C n f=(z,w):X Cⁿ⁺¹× C^n satisfying | w ( x ) | > h ( | z ( x ) | ) |w(x)|>h(|z(x)|) for all x ∈<!-- ∈ --> X x∈ X . In particular, f f may be chosen such that its limit set at infinity is a linearly embedded copy of C P n CP^n in C P 2 n CP²ⁿ .