Rawnsley's ε-Function on a Class of Bounded Hartogs Domains and its Applications

In this paper, by using the hypergeometric functions, we obtain the formula for the Rawnsley's epsilon-function of the Kahler manifold (H-{ki},gamma(n),g(mu,nu)) with mu is an element of(R+)(l )and nu is an element of (R+)(n-k), where H-{ki},gamma(n) is a class of bounded Hartogs domains defined by H-{ki},gamma(n):={z is an element of C-n:max(1 <= i <= l )& Vert;(z) over tilde(i)& Vert; < |z(k+1)|(gamma )< ... < |z(n)|gamma<1} and g(mu,nu) isa Kahlermetric associated with the Kahler potential -& sum;(l)(i)(=1)mu(i )ln (|z(k+1)|(2 gamma)- & Vert;(z) over tilde(i)& Vert;(2)) - & sum;(n )(j=k+1)nu(j )ln (|z(j+1)|(2)-|z(j)|(2)). As applications of the main result, we obtainthe existence of balanced metrics onH({ki},gamma)(n)and prove that H-{ki},gamma(n) admits a Berezin quantization.