Characterization of the Hilbert ball by its automorphism group (Q2782664)

In [Invent. Math. 41, 253-257 (1977; Zbl 0385.32016)] \textit{B. Wong} proved that a strongly pseudoconvex domain in \(\mathbb C^n\) with non-compact automorphism group must be the unit ball. In the intervening years there has been considerable work on this idea. \textit{J.-P. Rosay} [Ann. Inst. Fourier 29, 91-97 (1979; Zbl 0402.32001)] proved that the domain can be allowed to be bounded, but arbitrary, and that one need only assume that some boundary orbit accumulation point of the automorphism group action be strongly pseudoconvex in order to obtain the same conclusion. \textit{R. E. Green} and \textit{S. G. Krantz} [Lect. Notes Math. 1268, 121-157 (1987; Zbl 0626.32023)] considered the case when the orbit accumulation point is only weakly pseudoconvex. All of the work to date has been in complex space of finite dimension. NEWLINENEWLINENEWLINEIn the present paper the authors explore the case where the domain is in an infinite-dimensional Hilbert space. The principal result of the paper (Theorem 3.1) is as follows: Let \(\Omega\) be a bounded convex domain in a separable Hilbert space \(\mathcal H\). Assume that \(\Omega\) admits a boundary point \(\mathbf{p}\in\partial\Omega\) at which (1) \(\partial\Omega\) is \(C^2\) smooth and strongly pseudoconvex in a neighborhood of \textbf{p}, and (2) there exists \(\mathbf{q}\in\Omega\) and \(f_j\in\text{Aut }(\Omega)\) (\(j=1,2,\dots\)) such that \(f_j(\mathbf{q})\) converges to \textbf{p} in norm as \(j\to\infty\). Then \(\Omega\) is biholomorphic to the unit ball \(\mathbb B=\{z\in\mathcal H: \|z\|<1\}\).NEWLINENEWLINENEWLINEAlthough this result is analogous to the result of Rosay, its proof requires some new arguments. Key ingredients of the proof involve a new localization argument using holomorphic peaking functions and a new ``normal families'' argument in the construction of the biholomorphic map.