On the Bergman metric of symmettric spaces
Andrea Loi, Matteo Palmieri
TL;DR
The paper investigates bounded domains $\Omega\subset\mathbb{C}^n$ whose Bergman metric is locally symmetric, i.e. $\nabla R^\Omega=0$, and proves two rigidity results: if $g_\Omega$ is complete then $\Omega$ is symmetric; if $\Omega$ is pseudoconvex then $\Omega$ is biholomorphic to a complement $\widetilde{\Omega}\setminus E$ of a pluripolar set inside a bounded symmetric domain $\widetilde{\Omega}$. The authors combine Hermitian symmetric space structure, Calabi's theory of Kähler immersions into $\mathbb{CP}^\infty$ and the diastasis hereditary property with Bergman-kernel extension techniques and $A^2$ function theory to derive a uniformization-type description. The results show how a local curvature condition on the Bergman metric rigidly determines the global complex-analytic geometry, connecting local symmetricity to global symmetry or to a global model given by a bounded symmetric domain modulo a pluripolar set. A further discussion outlines possible generalizations to Stein manifolds and poses conjectures toward removing the pseudoconvexity hypothesis.
Abstract
We study bounded domains $Ω\subset\mathbb{C}^n$ whose Bergman metric is locally symmetric, i.e. its Riemannian curvature tensor is parallel with respect to the Levi-Civita connection. Following the strategy developed in \cite{UnifThm2}, we obtain two rigidity results. If the Bergman metric of $Ω$ is complete, then $Ω$ is (globally) symmetric. If instead $Ω$ is pseudoconvex, then $Ω$ is biholomorphic to $\widetildeΩ\setminus E$, where $\widetildeΩ\subset\mathbb{C}^n$ is a bounded symmetric domain and $E\subset\widetildeΩ$ is relatively closed and pluripolar. The proofs combine the structure theory of Hermitian symmetric spaces with Calabi's theory of Kähler immersions into the infinite dimensional complex projective space (in particular, rigidity and the hereditary property of the diastasis), together with analytic and pluripotential tools based on extension properties of square-integrable holomorphic functions and the Bergman kernel.