A Laplace integral on a Kähler manifold and Calabi's diastasis function
If \(M\) is a complex manifold endowed with a real analytic Kähler metric \(g\), and \(\alpha>1\), the Laplace integral of a smooth function \(f\) is defined by \[ L_\alpha(f)=\int_U f(y)\exp\bigl(-\alpha D(\cdot,y)\bigr)\omega^n(y)/n!, \] where \(\omega\) is the Kähler form of \(g\) and \(D\) is Calabi's diastasis function, defined on a suitable open subset \(U\times U\) of \(M\times M\). In this paper the author, using techniques due to \textit{M. Cahen, S. Gutt} and \textit{J. Rawnsley} [Trans. Am. Math. Soc. 337, No. 1, 73--98 (1993; Zbl 0788.53062)] generalizes results of \textit{M. Engliš} [J. Funct. Anal. 177, No. 2, 257--281 (2000; Zbl 0969.32003)] on the asymptotic expansion of \(L_\alpha(f)\) as \(\alpha\) goes to infinity.
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