On the Asymptotic Expansions of the Proper Harmonic Maps Between Balls in Bergman Metrics

Abstract

Let $$B_n$$ B n be the unit ball in "Equation missing" with the Bergman metric g and h is the Bergman metric on $$B_m$$ B m . Let $$u: (B_n, g)\rightarrow (B_m, h)$$ u : ( B n , g ) → ( B m , h ) be any harmonic map with $$\phi _0=u|_{\partial B_n}\in C^\infty (\partial B_n, \partial B_m)$$ ϕ 0 = u | ∂ B n ∈ C ∞ ( ∂ B n , ∂ B m ) . In this paper, we provide an asymptotic expansion formula for the above harmonic map u for a large class of $$\phi _0\in C^\infty (\partial B_n, \partial B_m)$$ ϕ 0 ∈ C ∞ ( ∂ B n , ∂ B m ) .