Bergman-harmonic maps of balls

We study Bergman-harmonic maps of balls $Phi :mathbb{B}_n o mathbb{B}_N$ extending either $C^2$ or $mathfrak{M}^1$ to the boundary of $mathbb{B}_n$. For every holomorphic (anti-holomorphic) map $Phi : mathbb{B}_n o mathbb{B}_N$ extending smoothly to the boundary we prove a Lichnerowicz type result i.e. we show that $E_{Omega_epsilon} (Psi ) geq E_{Omega_epsilon} (Phi ) + O(epsilon^{- n + 1})$ provided that $Psi$ is homotopic to $Phi$. When $Phi$ is proper, Bergman-harmonic, and $C^2$ up to the boundary, the boundary values map $phi : S^{2n-1} o S^{2N-1}$ is shown to satisfy a compatibility system similar to the tangential Cauchy-Riemann equations on $S^{2n-1}$ (and satisfied by the boundary values of any proper holomorphic map). For every weakly Bergman-harmonic map $Phi in W^1 (mathbb{B}_n , mathbb{B}_N )$ admitting Sobolev boundary values $phi in mathfrak{M}^1 (S^{2n-1} , mathbb{B}_N)$ the boundary values $phi$ are shown to be a weakly subelliptic harmonic map of $(S^{2n-1} , eta )$ into $(mathbb{B}_N , h)$, provided that $Phi^{-1} abla^h$ stays bounded at the boundary of $mathbb{B}_n$ and $phi$ has vanishing weak normal derivatives.