Extension of germs of holomorphic isometries up to normalizing constants with respect to the Bergman metric

Abstract

We study the extension problem for germs of holomorphic isometries $f:(D;x_0)\to (\Omega ;f(x_0))$ up to normalizing constants between bounded domains in Euclidean spaces equipped with Bergman metrics $d{s}_D^2$ on $D$ and $d{s}_{\Omega }^2$ on $\Omega$. Our main focus is on boundary extension for pairs of bounded domains $(D,\Omega )$ such that the Bergman kernel $K_D(z,w)$ extends meromorphically in $(z,\overline{w})$ to a neighborhood of $\overline{D}\times D$, and such that the analogous statement holds true for the Bergman kernel $K_{\Omega }(\zeta ,\xi )$ on $\Omega$. Assuming that $(D;d{s}_D^2)$ and $(\Omega ;d{s}_{\Omega }^2)$ are complete Kähler manifolds, we prove that the germ of map $f$ extends to a proper holomorphic isometric embedding such that Graph$(f)$ extends to a complex-analytic subvariety on some neigborhood of $\overline{D}\times \overline{\Omega }$. In the event that the Bergman kernel $K_D(z,w)$ extends to a rational function in $(z;\overline{w})$ and the analogue holds true for the Bergman kernel $K_{\Omega }(\zeta ,\xi )$, we show that Graph$(f)$ extends to an affine-algebraic variety. Our results apply especially to pairs $(D,\Omega )$ of bounded symmetric domains in their Harish-Chandra realizations. When $D$ is the complex unit ball $B^n$ of dimension $n\ge 2$, we obtain a new rigidity result which guarantees the total geodesy of the map under certain conditions. On the other hand, we construct examples of holomorphic isometries of the unit disk into polydisks which are not totally geodesic, answering in the negative a conjecture of Clozel-Ullmo's.