LIOUVILLE AND CALABI-YAU TYPE THEOREMS FOR COMPLEX HESSIAN EQUATIONS - Dinew, Slawomir; Kołodziej, Slawomir (American journal of mathematics, 2017)
We prove a Liouville type theorem for entire maximal m-subharmonic functions in ℂn with bounded gradient. This result, coupled with a standard blow-up argument, yields a (nonexplicit) a priori gradient estimate for the complex Hessian equation on a compact Kähler manifold. This terminates the program, initiated by Hou, Ma, and Wu, of solving the non-degenerate Hessian equation on such manifolds in full generality. We also obtain, using our previous work, continuous weak solutions in the degenerate case for the right-hand side in some Lp, with a sharp bound on p.