Liouville and Calabi-Yau type theorems for complex Hessian equations

Abstract

We prove a Liouville type theorem for entire maximal m-subharmonic functions in 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 Kähler manifold. This terminates the program, initiated in [HMW], 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 a sharp bound on p.

The Parabolic Flows for Complex Quotient Equations

“…Proof. The argument is very similar to those of Dinew and Kolodziej [9] and Gill [14], so we just give a brief statement here. We shall prove the theorem by contradiction and suppose that the gradient estimate (5.1) does not hold.…”

A variational approach to complex Hessian equations in Cn

“…As suggested by these authors, this estimate can be used in some blow-up analysis. The latter reduces the problem of solving equation (1.2) to a Liouville-type theorem for m-subharmonic functions in C n which was recently proved by Dinew and Kołodziej [11] and the solvability of equation (1.2) is thus confirmed on any compact Kähler manifold.…”

On a Class of Fully Nonlinear Elliptic Equations on Closed Hermitian Manifolds II: L∞ Estimate

“…By a result of Błocki [3], v is a maximal k-subharmonic function in C n . Now we can apply the Liouville theorem in [7] and find that v is a constant, which contradicts the construction of v.…”

“…In this section, we adapt the blowup method of Dinew and Kolodziej [7] to obtain the gradient estimate. Székelyhid [17] extended substantially the generality of the blowup method.…”