Upper boundary points of the gap intervals for rational maps between balls (Q1654923)

Let \(\mathbb B^n\) stand for the unit ball in \(\mathbb C^n\) and let \(\mathrm{Rat}(\mathbb B^n,\mathbb B^N)\) denote the set of all proper holomorphic rational maps from \(\mathbb B^n\) to \(\mathbb B^N\). We say that \(f,g\in\mathrm{Rat}(\mathbb B^n,\mathbb B^N)\) are equivalent if there are \(\sigma\in\mathrm{Aut}(\mathbb B^n)\) and \(\tau\in\mathrm{Aut}(\mathbb B^N)\) such that \(f=\tau\circ g\circ\sigma\). Put \[ \mathcal I_k:=(kn,(k+1)n-k(k+1)/2),\quad k=1,2,\dots,K(n):=\max\{t\in\mathbb N:t(t+1)/2>n\}. \] The gap conjecture claims that \(F\in \mathrm{Rat}(\mathbb B^n,\mathbb B^N)\), \(n\geq 3\), is equivalent to \((G,0')\), where \(G\in \mathrm{Rat}(\mathbb B^n,\mathbb B^{N'})\), \(N'<N\), if and only if \(N\in\mathcal I_k\) for some \(1\leq k\leq K(n)\). If \(N\) is a boundary point of the interval \(\mathcal I_1\) or \(\mathcal I_2\), the maps from \(Rat(\mathbb B^n,\mathbb B^N)\) already have been determined. In the paper under review the authors prove the following result. Let \(F\in \mathrm{Rat}(\mathbb B^n,\mathbb B^N)\), \(n\geq3\), and let \(N\) be the upper boundary point of the interval \(\mathcal I_k\), \(k\leq n-2\). If the geometric rank of \(F\) is \(k\), then \(F\) is equivalent to the generalized Whitney map \(W_{n,k}\).