Eversive maps of bounded convex domains in \(\mathbb{C}^{n+1}\) (Q1407748)

An automorphism group \(G = \Aut(B^{n+1})\) of the unit ball \(B^{n+1}\subset {\mathbb C}^{n+1}\) has the following two important properties: it has the largest admissible dimension for an automorphism group of an Hermitian manifold and it acts freely and transitively on the bundle \(\pi : U(B^{n+1})\rightarrow B^{n+1}\) of all unitary frames of \(B^{n+1}\) with respect to the Poincaré-Bergmann metric. Each element \(g \in G\) extends up to boundary \(S^{n+1} = \partial B^{n+1}\) and it acts as a CR transformation on \(S^{2n+1}.\) Moreover, \(G\) has the largest admissible dimension for an automorphism group of a Levi non-degenerate hypersurface and it acts freely and transitively on the Chern-Moser bundle \(\pi': P_{\text{CM}}(S^{2n+1}) \rightarrow S^{2n+1},\) associated with the CR structure of \(S^{2n+1}.\) It follows that both manifolds \(U(B^{n+1})\) and \(P_{\text{CM}}(S^{2n+1})\) are diffeomorphic to \(G.\) Therefore, given a point \(u \in U(B^{n+1})\) and a point in \(p \in P_{\text{CM}}(S^{2n+1}),\) there exists a unique \(G\)-equivariant diffeomorphism \(E: U(B^{n+1}) \rightarrow P_{\text{CM}} (S^{2n+1})\) mapping \(u\) into \(p.\) The associated diffeomorphism determines a quite interesting duality between the invariants of the Hermitian geometry of \(B^{n+1}\) and the invariants of the CR geometry of the unit sphere \(S^{2n+1}.\) Such duality constitutes the perfect analog of the well-known relation between the geometry of the hyperbolic space \(H^{n+1}\) and the conformal geometry of unit sphere \(S^{n}.\) The main goal of this article is to prove that one diffeomorphism as above exists for any smoothly bounded, strongly convex domain \(D \subset {\mathbb C}^{n+1}.\) A duality principle, relating the geometry of the Kobayashi metric with the CR geometry of the boundaries of smoothly bounded, strongly convex domain \(D \subset {\mathbb C}^{n+1},\) is established. As a by-product, the authors are able to show the following new fact: any holomorphic Jacobi vector field of a smoothly bounded, strongly convex domain \(D\) is uniquely determined by its second order jet at a boundary point \(y \in \partial D.\)