THE HEAT KERNEL OF A BALL IN C~n

By introducing the horosphere coordinate of a unit ball B~n in C~n and an integraltransformation formula of functions in such coordidates,the author constructs the heatkernel H_(B~n)(z,w,t)of the heat equation associated to the Bergman metric of B~n.That iswhere c_n is a well-defined constant and r(z,w)is the geodesic destance of two points zand w of B_n and t∈ R~+.Sincethenis the Green function of the topological product space B~m×B~n.

The Heat Kernel of a Ball in C^n

By introducing the horosphere coordinate of a unit ball B^n in C^n and an integral transformation formula of functions in such coordidates, the author constructs the heat kernel H_B^n(z,w,t) of the heat equation associated to the Bergman metric of B^n.That is $H_B^n(z,w,t)=c_n(-1/\pi)^ne^{-n^2t}/\sqrt(t)\int_-\infty^\infty{1/sh2\sigma\partial/\partial\sigma(1/sh\sigma\partial/\partial)^n-1e^{-\sigma^2/4t}]_{ch2\sigma=ch2r(x,w)+\tau^2}d\tau$ where c_n is a well-defined constant and r(z, w) is the geodesic destanco of two points s and w of B^n and t\in R^+. Since $H_B^m*B^n=H_B^m\cdot H_B^n$ then $G((z_1,z_2),(w_1,w_2))=-\int_0^\infty{H_B^m(z_1,w_1,t)H_B^n(z_2,w_2,t)}dt$ is the Green function of the topological product space B^m*B^n.