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

Let B n be the unit ball in with the Bergman metric g and h is the Bergman metric on B m . Let u : ( B n , g ) → ( B m , h ) be any harmonic map with 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 ϕ 0 ∈ C ∞ ( ∂ B n , ∂ B m ) .