Liouville and Calabi-Yau type theorems for complex Hessian equations

Details Liouville and Calabi-Yau type theorems for complex Hessian equations Liouville and Calabi-Yau type theorems for complex Hessian equations

2012-03-18

arxiv.org/abs/1203.3995v1

Mathematics

Complex Variables

10 pages

Scientific paper

We prove a Liouville type theorem for entire maximal $m$-subharmonic functions in $\mathbb C^n$ with bounded gradient. This result, coupled with a standard blow-up argument, yields a (non-explicit) a priori gradient estimate for the complex Hessian equation on a compact K\"ahler 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 $L^p, $ with sharp bound on $p$.

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