Proposition

Corollary 1. For every ε>0 and 1<c<9/4 , all sufficiently large integers k admit metrics with the conclusions of Theorem 1.1 and (1+ε)^ -2 g_ E ≤g_k ≤(1+ε)^2g_ E ℝ^3. Their asymptotic volume ratio can be prescribed arbitrarily close to 1 . …

Corollary 1. For every ε>0 and 1<c<9/4 , all sufficiently large integers k admit metrics with the conclusions of Theorem 1.1 and (1+ε)^-2 g_E ≤g_k ≤(1+ε)^2g_ E ℝ^3. Their asymptotic volume ratio can be prescribed arbitrarily close to 1 . In particular, the identity map is globally (1+ε) -bi-Lipschitz.