A Kähler potential on the unit ball with constant differential norm

Abstract

Let Bn be the unit ball in Cn and Hn be the homogeneous Siegel domain of the second kind which is biholomorphic to Bn . We show that the Kähler potential of Hn is unique up to the automorphisms among Kähler potentials whose differentials have constant norms. As an application, we consider a domain Ω in Cn , which is biholomorphic to Bn . We show that if Ω is affine homogeneous, then it is affine equivalent to Hn . Assume next that its canonical potential with respect to the Kähler–Einstein metric has a differential with a constant norm. If the biholomorphism between Ω and Bn is a restriction of a Möbius transformation, then the map is affine equivalent to a Cayley transform. © 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.