An analogue of minimal surface theory in SL(n,C)/SU(n)
  We shall discuss the class of surfaces with holomorphic right Gauss maps in non-compact duals of compact semisimple Lie groups (e.g. SL(n,C)/SU(n)), which contains minimal surfaces in R^n and constant mean curvature 1 surfaces in H^3. A Weierstrass type representation formula, and a Chern-Osserman type inequality for such surfaces are given. 
Masatoshi Kokubu, Masaro Takahashi, Masaaki Umehara, Kotaro Yamada